{"id":65455,"date":"2026-08-17T15:37:00","date_gmt":"2026-08-17T14:37:00","guid":{"rendered":"https:\/\/blog-admin.thethinkacademy.com\/?p=65455"},"modified":"2026-08-17T15:37:02","modified_gmt":"2026-08-17T14:37:02","slug":"rules-in-divisibility-guide","status":"publish","type":"post","link":"https:\/\/blog-admin.thethinkacademy.com\/blog\/2026\/08\/17\/rules-in-divisibility-guide\/","title":{"rendered":"Rules of Divisibility Explained: Complete Guide with Examples"},"content":{"rendered":"\n<p class=\"wp-block-paragraph\">The rules in divisibility are a set of shortcuts that tell you whether one number divides another without performing full division. They are foundational number theory \u2014 taught from Grade 5 onward, tested on EQAO, and a standard time-saving tool in every major mathematics competition from the AMC 8 through to the Euclid Contest. Knowing the rules in divisibility automatically is one of the simplest ways a student can save time and reduce errors on timed assessments.<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">This guide covers every major rule of divisibility with clear explanations, proofs of why each rule works, worked examples, and practice problems.<\/p>\n\n\n\n<hr class=\"wp-block-separator has-alpha-channel-opacity\"\/>\n\n\n\n<h2 class=\"wp-block-heading\">What Are the Rules in Divisibility?<\/h2>\n\n\n\n<p class=\"wp-block-paragraph\">A <strong>divisibility rule<\/strong> is a shortcut for determining whether an integer is divisible by a given number without performing long division. The rules in divisibility exploit specific properties of the number system \u2014 usually properties of the digits in base 10.<\/p>\n\n\n\n<p class=\"wp-block-paragraph\"><strong>Why they matter:<\/strong> The rules in divisibility are not just shortcuts for calculation \u2014 they reveal structure in the number system. Understanding why the rule for 3 works (digit sum divisible by 3) is the beginning of understanding modular arithmetic, which is one of the most important tools in number theory at every competition level.<\/p>\n\n\n\n<hr class=\"wp-block-separator has-alpha-channel-opacity\"\/>\n\n\n\n<h2 class=\"wp-block-heading\">Rules in Divisibility for 2<\/h2>\n\n\n\n<p class=\"wp-block-paragraph\"><strong>Rule:<\/strong> A number is divisible by 2 if its last digit is even (0, 2, 4, 6, or 8).<\/p>\n\n\n\n<p class=\"wp-block-paragraph\"><strong>Examples:<\/strong><\/p>\n\n\n\n<ul class=\"wp-block-list\">\n<li>4,836 \u2192 last digit 6 (even) \u2192 <strong>divisible by 2<\/strong> \u2713<\/li>\n\n\n\n<li>7,293 \u2192 last digit 3 (odd) \u2192 <strong>not divisible by 2<\/strong> \u2717<\/li>\n<\/ul>\n\n\n\n<p class=\"wp-block-paragraph\"><strong>Why it works:<\/strong> Any number can be written as 10q + r where r is the last digit. Since 10 is divisible by 2, whether the whole number is divisible by 2 depends entirely on whether r is. And r is even (0, 2, 4, 6, 8) if and only if 2 divides r.<\/p>\n\n\n\n<hr class=\"wp-block-separator has-alpha-channel-opacity\"\/>\n\n\n\n<h2 class=\"wp-block-heading\">Rules in Divisibility for 3<\/h2>\n\n\n\n<p class=\"wp-block-paragraph\"><strong>Rule:<\/strong> A number is divisible by 3 if the sum of its digits is divisible by 3.<\/p>\n\n\n\n<p class=\"wp-block-paragraph\"><strong>Examples:<\/strong><\/p>\n\n\n\n<ul class=\"wp-block-list\">\n<li>2,184 \u2192 digits 2+1+8+4 = 15 \u2192 15 \u00f7 3 = 5 \u2192 <strong>divisible by 3<\/strong> \u2713<\/li>\n\n\n\n<li>7,421 \u2192 digits 7+4+2+1 = 14 \u2192 14 is not divisible by 3 \u2192 <strong>not divisible by 3<\/strong> \u2717<\/li>\n<\/ul>\n\n\n\n<p class=\"wp-block-paragraph\"><strong>Why it works:<\/strong> Any number <math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mi>N<\/mi><\/mrow><annotation encoding=\"application\/x-tex\">N<\/annotation><\/semantics><\/math>N with digits <math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><msub><mi>d<\/mi><mi>n<\/mi><\/msub><msub><mi>d<\/mi><mrow><mi>n<\/mi><mo>\u2212<\/mo><mn>1<\/mn><\/mrow><\/msub><mo>\u2026<\/mo><msub><mi>d<\/mi><mn>1<\/mn><\/msub><msub><mi>d<\/mi><mn>0<\/mn><\/msub><\/mrow><annotation encoding=\"application\/x-tex\">d_n d_{n-1} \\ldots d_1 d_0<\/annotation><\/semantics><\/math>dn\u200bdn\u22121\u200b\u2026d1\u200bd0\u200b can be written as:<math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" display=\"block\"><semantics><mrow><mi>N<\/mi><mo>=<\/mo><msub><mi>d<\/mi><mi>n<\/mi><\/msub><mo>\u22c5<\/mo><msup><mn>10<\/mn><mi>n<\/mi><\/msup><mo>+<\/mo><msub><mi>d<\/mi><mrow><mi>n<\/mi><mo>\u2212<\/mo><mn>1<\/mn><\/mrow><\/msub><mo>\u22c5<\/mo><msup><mn>10<\/mn><mrow><mi>n<\/mi><mo>\u2212<\/mo><mn>1<\/mn><\/mrow><\/msup><mo>+<\/mo><mo>\u22ef<\/mo><mo>+<\/mo><msub><mi>d<\/mi><mn>1<\/mn><\/msub><mo>\u22c5<\/mo><mn>10<\/mn><mo>+<\/mo><msub><mi>d<\/mi><mn>0<\/mn><\/msub><\/mrow><annotation encoding=\"application\/x-tex\">N = d_n \\cdot 10^n + d_{n-1} \\cdot 10^{n-1} + \\cdots + d_1 \\cdot 10 + d_0<\/annotation><\/semantics><\/math>N=dn\u200b\u22c510n+dn\u22121\u200b\u22c510n\u22121+\u22ef+d1\u200b\u22c510+d0\u200b<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">Since <math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mn>10<\/mn><mo>\u2261<\/mo><mn>1<\/mn><mspace><\/mspace><mspace width=\"0.4444em\"><\/mspace><mo stretchy=\"false\">(<\/mo><mrow><mi mathvariant=\"normal\">m<\/mi><mi mathvariant=\"normal\">o<\/mi><mi mathvariant=\"normal\">d<\/mi><\/mrow><mspace width=\"0.3333em\"><\/mspace><mn>3<\/mn><mo stretchy=\"false\">)<\/mo><\/mrow><annotation encoding=\"application\/x-tex\">10 \\equiv 1 \\pmod{3}<\/annotation><\/semantics><\/math>10\u22611(mod3), every power of 10 is <math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mo>\u2261<\/mo><mn>1<\/mn><mspace><\/mspace><mspace width=\"0.4444em\"><\/mspace><mo stretchy=\"false\">(<\/mo><mrow><mi mathvariant=\"normal\">m<\/mi><mi mathvariant=\"normal\">o<\/mi><mi mathvariant=\"normal\">d<\/mi><\/mrow><mspace width=\"0.3333em\"><\/mspace><mn>3<\/mn><mo stretchy=\"false\">)<\/mo><\/mrow><annotation encoding=\"application\/x-tex\">\\equiv 1 \\pmod{3}<\/annotation><\/semantics><\/math>\u22611(mod3). Therefore:<math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" display=\"block\"><semantics><mrow><mi>N<\/mi><mo>\u2261<\/mo><msub><mi>d<\/mi><mi>n<\/mi><\/msub><mo>+<\/mo><msub><mi>d<\/mi><mrow><mi>n<\/mi><mo>\u2212<\/mo><mn>1<\/mn><\/mrow><\/msub><mo>+<\/mo><mo>\u22ef<\/mo><mo>+<\/mo><msub><mi>d<\/mi><mn>1<\/mn><\/msub><mo>+<\/mo><msub><mi>d<\/mi><mn>0<\/mn><\/msub><mspace><\/mspace><mspace width=\"1em\"><\/mspace><mo stretchy=\"false\">(<\/mo><mrow><mi mathvariant=\"normal\">m<\/mi><mi mathvariant=\"normal\">o<\/mi><mi mathvariant=\"normal\">d<\/mi><\/mrow><mspace width=\"0.3333em\"><\/mspace><mn>3<\/mn><mo stretchy=\"false\">)<\/mo><\/mrow><annotation encoding=\"application\/x-tex\">N \\equiv d_n + d_{n-1} + \\cdots + d_1 + d_0 \\pmod{3}<\/annotation><\/semantics><\/math>N\u2261dn\u200b+dn\u22121\u200b+\u22ef+d1\u200b+d0\u200b(mod3)<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">So <math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mn>3<\/mn><mo>\u2223<\/mo><mi>N<\/mi><\/mrow><annotation encoding=\"application\/x-tex\">3 \\mid N<\/annotation><\/semantics><\/math>3\u2223N if and only if <math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mn>3<\/mn><mo>\u2223<\/mo><mo stretchy=\"false\">(<\/mo><msub><mi>d<\/mi><mi>n<\/mi><\/msub><mo>+<\/mo><msub><mi>d<\/mi><mrow><mi>n<\/mi><mo>\u2212<\/mo><mn>1<\/mn><\/mrow><\/msub><mo>+<\/mo><mo>\u22ef<\/mo><mo>+<\/mo><msub><mi>d<\/mi><mn>0<\/mn><\/msub><mo stretchy=\"false\">)<\/mo><\/mrow><annotation encoding=\"application\/x-tex\">3 \\mid (d_n + d_{n-1} + \\cdots + d_0)<\/annotation><\/semantics><\/math>3\u2223(dn\u200b+dn\u22121\u200b+\u22ef+d0\u200b). \u25a1<\/p>\n\n\n\n<hr class=\"wp-block-separator has-alpha-channel-opacity\"\/>\n\n\n\n<h2 class=\"wp-block-heading\">Divisibility Rule for 4<\/h2>\n\n\n\n<p class=\"wp-block-paragraph\"><strong>Rule:<\/strong> A number is divisible by 4 if its last two digits form a number divisible by 4.<\/p>\n\n\n\n<p class=\"wp-block-paragraph\"><strong>Examples:<\/strong><\/p>\n\n\n\n<ul class=\"wp-block-list\">\n<li>2,184 \u2192 last two digits 84 \u2192 84 \u00f7 4 = 21 \u2192 <strong>divisible by 4<\/strong> \u2713<\/li>\n\n\n\n<li>6,730 \u2192 last two digits 30 \u2192 30 \u00f7 4 = 7.5 \u2192 <strong>not divisible by 4<\/strong> \u2717<\/li>\n<\/ul>\n\n\n\n<p class=\"wp-block-paragraph\"><strong>Why it works:<\/strong> Any number can be written as 100q + r where r is the two-digit number formed by the last two digits. Since 100 = 4 \u00d7 25 is divisible by 4, the divisibility of the whole number by 4 depends only on r.<\/p>\n\n\n\n<hr class=\"wp-block-separator has-alpha-channel-opacity\"\/>\n\n\n\n<h2 class=\"wp-block-heading\">Divisibility Rule for 5<\/h2>\n\n\n\n<p class=\"wp-block-paragraph\"><strong>Rule:<\/strong> A number is divisible by 5 if its last digit is 0 or 5.<\/p>\n\n\n\n<p class=\"wp-block-paragraph\"><strong>Examples:<\/strong><\/p>\n\n\n\n<ul class=\"wp-block-list\">\n<li>3,475 \u2192 last digit 5 \u2192 <strong>divisible by 5<\/strong> \u2713<\/li>\n\n\n\n<li>8,242 \u2192 last digit 2 \u2192 <strong>not divisible by 5<\/strong> \u2717<\/li>\n<\/ul>\n\n\n\n<p class=\"wp-block-paragraph\"><strong>Why it works:<\/strong> Any number is 10q + r where r is the last digit. Since 10 = 5 \u00d7 2 is divisible by 5, divisibility depends entirely on whether 5 divides r. The only single-digit multiples of 5 are 0 and 5.<\/p>\n\n\n\n<hr class=\"wp-block-separator has-alpha-channel-opacity\"\/>\n\n\n\n<h2 class=\"wp-block-heading\">Divisibility Rule for 6<\/h2>\n\n\n\n<p class=\"wp-block-paragraph\"><strong>Rule:<\/strong> A number is divisible by 6 if it is divisible by both 2 AND 3.<\/p>\n\n\n\n<p class=\"wp-block-paragraph\"><strong>Examples:<\/strong><\/p>\n\n\n\n<ul class=\"wp-block-list\">\n<li>2,184 \u2192 even (\u00f72 \u2713) AND digit sum 15 (\u00f73 \u2713) \u2192 <strong>divisible by 6<\/strong> \u2713<\/li>\n\n\n\n<li>5,310 \u2192 even (\u00f72 \u2713) but digit sum 9&#8230; wait, 5+3+1+0=9 \u00f73 \u2713 \u2192 <strong>divisible by 6<\/strong> \u2713<\/li>\n\n\n\n<li>4,514 \u2192 even (\u00f72 \u2713) but digit sum 14, not \u00f73 \u2192 <strong>not divisible by 6<\/strong> \u2717<\/li>\n<\/ul>\n\n\n\n<p class=\"wp-block-paragraph\"><strong>Why it works:<\/strong> <math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mn>6<\/mn><mo>=<\/mo><mn>2<\/mn><mo>\u00d7<\/mo><mn>3<\/mn><\/mrow><annotation encoding=\"application\/x-tex\">6 = 2 \\times 3<\/annotation><\/semantics><\/math>6=2\u00d73 and <math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mi>gcd<\/mi><mo>\u2061<\/mo><mo stretchy=\"false\">(<\/mo><mn>2<\/mn><mo separator=\"true\">,<\/mo><mn>3<\/mn><mo stretchy=\"false\">)<\/mo><mo>=<\/mo><mn>1<\/mn><\/mrow><annotation encoding=\"application\/x-tex\">\\gcd(2,3) = 1<\/annotation><\/semantics><\/math>gcd(2,3)=1. When two coprime numbers both divide <math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mi>n<\/mi><\/mrow><annotation encoding=\"application\/x-tex\">n<\/annotation><\/semantics><\/math>n, their product also divides <math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mi>n<\/mi><\/mrow><annotation encoding=\"application\/x-tex\">n<\/annotation><\/semantics><\/math>n. Since 2 and 3 are coprime, divisibility by 6 is exactly divisibility by both 2 and 3.<\/p>\n\n\n\n<p class=\"wp-block-paragraph\"><strong>Key insight:<\/strong> This approach generalises. Divisibility by 15 = 3 \u00d7 5 requires divisibility by both 3 and 5. Divisibility by 12 = 4 \u00d7 3 requires divisibility by both 4 and 3 (since <math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mi>gcd<\/mi><mo>\u2061<\/mo><mo stretchy=\"false\">(<\/mo><mn>4<\/mn><mo separator=\"true\">,<\/mo><mn>3<\/mn><mo stretchy=\"false\">)<\/mo><mo>=<\/mo><mn>1<\/mn><\/mrow><annotation encoding=\"application\/x-tex\">\\gcd(4,3) = 1<\/annotation><\/semantics><\/math>gcd(4,3)=1). Divisibility by 6 is the simplest example of this composite rule.<\/p>\n\n\n\n<figure class=\"wp-block-image size-large\"><a href=\"https:\/\/www.thinkacademy.ca\/free-assessment?source_id=6721&amp;source_type=9&amp;utm_medium=website&amp;utm_source=pc_blog\"><img loading=\"lazy\" decoding=\"async\" width=\"1024\" height=\"409\" src=\"https:\/\/blog-admin.thethinkacademy.com\/wp-content\/uploads\/2026\/08\/ChatGPT-Image-Aug-17-2026-03_25_41-PM-1-1024x409.png\" alt=\"Rules in Divisibility cta\" class=\"wp-image-65457\" srcset=\"https:\/\/blog-admin.thethinkacademy.com\/wp-content\/uploads\/2026\/08\/ChatGPT-Image-Aug-17-2026-03_25_41-PM-1-1024x409.png 1024w, https:\/\/blog-admin.thethinkacademy.com\/wp-content\/uploads\/2026\/08\/ChatGPT-Image-Aug-17-2026-03_25_41-PM-1-300x120.png 300w, https:\/\/blog-admin.thethinkacademy.com\/wp-content\/uploads\/2026\/08\/ChatGPT-Image-Aug-17-2026-03_25_41-PM-1-768x307.png 768w, https:\/\/blog-admin.thethinkacademy.com\/wp-content\/uploads\/2026\/08\/ChatGPT-Image-Aug-17-2026-03_25_41-PM-1-1536x614.png 1536w, https:\/\/blog-admin.thethinkacademy.com\/wp-content\/uploads\/2026\/08\/ChatGPT-Image-Aug-17-2026-03_25_41-PM-1-1300x520.png 1300w, https:\/\/blog-admin.thethinkacademy.com\/wp-content\/uploads\/2026\/08\/ChatGPT-Image-Aug-17-2026-03_25_41-PM-1.png 1983w\" sizes=\"auto, (max-width: 1024px) 100vw, 1024px\" \/><\/a><\/figure>\n\n\n\n<hr class=\"wp-block-separator has-alpha-channel-opacity\"\/>\n\n\n\n<h2 class=\"wp-block-heading\">Divisibility Rule for 7<\/h2>\n\n\n\n<p class=\"wp-block-paragraph\"><strong>Rule:<\/strong> Double the last digit and subtract from the remaining number. If the result is divisible by 7 (including 0), the original is divisible by 7.<\/p>\n\n\n\n<p class=\"wp-block-paragraph\"><strong>Repeat as necessary for large numbers.<\/strong><\/p>\n\n\n\n<p class=\"wp-block-paragraph\"><strong>Example:<\/strong> Is 2,184 divisible by 7?<\/p>\n\n\n\n<ul class=\"wp-block-list\">\n<li>Last digit: 4. Remaining: 218. Double 4: 8.<\/li>\n\n\n\n<li>218 \u2212 8 = 210.<\/li>\n\n\n\n<li>Is 210 divisible by 7? 210 \u00f7 7 = 30. <strong>Yes.<\/strong><\/li>\n\n\n\n<li>Therefore <strong>2,184 is divisible by 7<\/strong> \u2713<\/li>\n<\/ul>\n\n\n\n<p class=\"wp-block-paragraph\">Check: 2,184 \u00f7 7 = 312. \u2713<\/p>\n\n\n\n<p class=\"wp-block-paragraph\"><strong>Another example:<\/strong> Is 5,831 divisible by 7?<\/p>\n\n\n\n<ul class=\"wp-block-list\">\n<li>Last digit 1, remaining 583. Subtract 2\u00d71=2: 583\u22122=581.<\/li>\n\n\n\n<li>Last digit 1, remaining 58. Subtract 2\u00d71=2: 58\u22122=56.<\/li>\n\n\n\n<li>56 \u00f7 7 = 8. <strong>Yes.<\/strong><\/li>\n\n\n\n<li><strong>5,831 is divisible by 7<\/strong> \u2713<\/li>\n<\/ul>\n\n\n\n<p class=\"wp-block-paragraph\"><strong>Why it works:<\/strong> <math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mn>10<\/mn><mo>\u2261<\/mo><mn>3<\/mn><mspace><\/mspace><mspace width=\"0.4444em\"><\/mspace><mo stretchy=\"false\">(<\/mo><mrow><mi mathvariant=\"normal\">m<\/mi><mi mathvariant=\"normal\">o<\/mi><mi mathvariant=\"normal\">d<\/mi><\/mrow><mspace width=\"0.3333em\"><\/mspace><mn>7<\/mn><mo stretchy=\"false\">)<\/mo><\/mrow><annotation encoding=\"application\/x-tex\">10 \\equiv 3 \\pmod{7}<\/annotation><\/semantics><\/math>10\u22613(mod7), and <math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mn>20<\/mn><mo>\u2261<\/mo><mn>6<\/mn><mo>\u2261<\/mo><mo>\u2212<\/mo><mn>1<\/mn><mspace><\/mspace><mspace width=\"0.4444em\"><\/mspace><mo stretchy=\"false\">(<\/mo><mrow><mi mathvariant=\"normal\">m<\/mi><mi mathvariant=\"normal\">o<\/mi><mi mathvariant=\"normal\">d<\/mi><\/mrow><mspace width=\"0.3333em\"><\/mspace><mn>7<\/mn><mo stretchy=\"false\">)<\/mo><\/mrow><annotation encoding=\"application\/x-tex\">20 \\equiv 6 \\equiv -1 \\pmod{7}<\/annotation><\/semantics><\/math>20\u22616\u2261\u22121(mod7). Writing <math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mi>N<\/mi><mo>=<\/mo><mn>10<\/mn><mi>q<\/mi><mo>+<\/mo><mi>d<\/mi><\/mrow><annotation encoding=\"application\/x-tex\">N = 10q + d<\/annotation><\/semantics><\/math>N=10q+d (last digit <math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mi>d<\/mi><\/mrow><annotation encoding=\"application\/x-tex\">d<\/annotation><\/semantics><\/math>d): <math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mi>N<\/mi><mo>\u2261<\/mo><mn>10<\/mn><mi>q<\/mi><mo>+<\/mo><mi>d<\/mi><mspace><\/mspace><mspace width=\"0.4444em\"><\/mspace><mo stretchy=\"false\">(<\/mo><mrow><mi mathvariant=\"normal\">m<\/mi><mi mathvariant=\"normal\">o<\/mi><mi mathvariant=\"normal\">d<\/mi><\/mrow><mspace width=\"0.3333em\"><\/mspace><mn>7<\/mn><mo stretchy=\"false\">)<\/mo><\/mrow><annotation encoding=\"application\/x-tex\">N \\equiv 10q + d \\pmod{7}<\/annotation><\/semantics><\/math>N\u226110q+d(mod7). We want to check whether <math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mi>q<\/mi><mo>\u2212<\/mo><mn>2<\/mn><mi>d<\/mi><mo>\u2261<\/mo><mn>0<\/mn><mspace><\/mspace><mspace width=\"0.4444em\"><\/mspace><mo stretchy=\"false\">(<\/mo><mrow><mi mathvariant=\"normal\">m<\/mi><mi mathvariant=\"normal\">o<\/mi><mi mathvariant=\"normal\">d<\/mi><\/mrow><mspace width=\"0.3333em\"><\/mspace><mn>7<\/mn><mo stretchy=\"false\">)<\/mo><\/mrow><annotation encoding=\"application\/x-tex\">q &#8211; 2d \\equiv 0 \\pmod 7<\/annotation><\/semantics><\/math>q\u22122d\u22610(mod7), which is equivalent since <math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mn>10<\/mn><mo stretchy=\"false\">(<\/mo><mi>q<\/mi><mo>\u2212<\/mo><mn>2<\/mn><mi>d<\/mi><mo stretchy=\"false\">)<\/mo><mo>=<\/mo><mn>10<\/mn><mi>q<\/mi><mo>\u2212<\/mo><mn>20<\/mn><mi>d<\/mi><mo>\u2261<\/mo><mn>10<\/mn><mi>q<\/mi><mo>+<\/mo><mi>d<\/mi><mo>\u2212<\/mo><mn>21<\/mn><mi>d<\/mi><mo>=<\/mo><mi>N<\/mi><mo>\u2212<\/mo><mn>21<\/mn><mi>d<\/mi><mo>\u2261<\/mo><mi>N<\/mi><mspace><\/mspace><mspace width=\"0.4444em\"><\/mspace><mo stretchy=\"false\">(<\/mo><mrow><mi mathvariant=\"normal\">m<\/mi><mi mathvariant=\"normal\">o<\/mi><mi mathvariant=\"normal\">d<\/mi><\/mrow><mspace width=\"0.3333em\"><\/mspace><mn>7<\/mn><mo stretchy=\"false\">)<\/mo><\/mrow><annotation encoding=\"application\/x-tex\">10(q &#8211; 2d) = 10q &#8211; 20d \\equiv 10q + d &#8211; 21d = N &#8211; 21d \\equiv N \\pmod{7}<\/annotation><\/semantics><\/math>10(q\u22122d)=10q\u221220d\u226110q+d\u221221d=N\u221221d\u2261N(mod7).<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">The rule for 7 is the least elegant \u2014 for most competition purposes, just dividing directly is faster than applying this rule. It is included for completeness and because the proof is instructive.<\/p>\n\n\n\n<hr class=\"wp-block-separator has-alpha-channel-opacity\"\/>\n\n\n\n<h2 class=\"wp-block-heading\">Divisibility Rule for 8<\/h2>\n\n\n\n<p class=\"wp-block-paragraph\"><strong>Rule:<\/strong> A number is divisible by 8 if its last three digits form a number divisible by 8.<\/p>\n\n\n\n<p class=\"wp-block-paragraph\"><strong>Examples:<\/strong><\/p>\n\n\n\n<ul class=\"wp-block-list\">\n<li>5,184 \u2192 last three digits 184 \u2192 184 \u00f7 8 = 23 \u2192 <strong>divisible by 8<\/strong> \u2713<\/li>\n\n\n\n<li>7,356 \u2192 last three digits 356 \u2192 356 \u00f7 8 = 44.5 \u2192 <strong>not divisible by 8<\/strong> \u2717<\/li>\n<\/ul>\n\n\n\n<p class=\"wp-block-paragraph\"><strong>Why it works:<\/strong> Any number can be written as 1000q + r where r is the last three digits. Since 1000 = 8 \u00d7 125 is divisible by 8, divisibility depends only on r.<\/p>\n\n\n\n<hr class=\"wp-block-separator has-alpha-channel-opacity\"\/>\n\n\n\n<h2 class=\"wp-block-heading\">Divisibility Rule for 9<\/h2>\n\n\n\n<p class=\"wp-block-paragraph\"><strong>Rule:<\/strong> A number is divisible by 9 if the sum of its digits is divisible by 9.<\/p>\n\n\n\n<p class=\"wp-block-paragraph\"><strong>Examples:<\/strong><\/p>\n\n\n\n<ul class=\"wp-block-list\">\n<li>3,456 \u2192 3+4+5+6 = 18 \u2192 18 \u00f7 9 = 2 \u2192 <strong>divisible by 9<\/strong> \u2713<\/li>\n\n\n\n<li>7,841 \u2192 7+8+4+1 = 20 \u2192 not divisible by 9 \u2192 <strong>not divisible by 9<\/strong> \u2717<\/li>\n<\/ul>\n\n\n\n<p class=\"wp-block-paragraph\"><strong>Why it works:<\/strong> Identical reasoning to the rule for 3, using <math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mn>10<\/mn><mo>\u2261<\/mo><mn>1<\/mn><mspace><\/mspace><mspace width=\"0.4444em\"><\/mspace><mo stretchy=\"false\">(<\/mo><mrow><mi mathvariant=\"normal\">m<\/mi><mi mathvariant=\"normal\">o<\/mi><mi mathvariant=\"normal\">d<\/mi><\/mrow><mspace width=\"0.3333em\"><\/mspace><mn>9<\/mn><mo stretchy=\"false\">)<\/mo><\/mrow><annotation encoding=\"application\/x-tex\">10 \\equiv 1 \\pmod{9}<\/annotation><\/semantics><\/math>10\u22611(mod9) instead of <math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mn>10<\/mn><mo>\u2261<\/mo><mn>1<\/mn><mspace><\/mspace><mspace width=\"0.4444em\"><\/mspace><mo stretchy=\"false\">(<\/mo><mrow><mi mathvariant=\"normal\">m<\/mi><mi mathvariant=\"normal\">o<\/mi><mi mathvariant=\"normal\">d<\/mi><\/mrow><mspace width=\"0.3333em\"><\/mspace><mn>3<\/mn><mo stretchy=\"false\">)<\/mo><\/mrow><annotation encoding=\"application\/x-tex\">10 \\equiv 1 \\pmod{3}<\/annotation><\/semantics><\/math>10\u22611(mod3).<math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" display=\"block\"><semantics><mrow><mi>N<\/mi><mo>\u2261<\/mo><msub><mi>d<\/mi><mi>n<\/mi><\/msub><mo>+<\/mo><msub><mi>d<\/mi><mrow><mi>n<\/mi><mo>\u2212<\/mo><mn>1<\/mn><\/mrow><\/msub><mo>+<\/mo><mo>\u22ef<\/mo><mo>+<\/mo><msub><mi>d<\/mi><mn>0<\/mn><\/msub><mspace><\/mspace><mspace width=\"1em\"><\/mspace><mo stretchy=\"false\">(<\/mo><mrow><mi mathvariant=\"normal\">m<\/mi><mi mathvariant=\"normal\">o<\/mi><mi mathvariant=\"normal\">d<\/mi><\/mrow><mspace width=\"0.3333em\"><\/mspace><mn>9<\/mn><mo stretchy=\"false\">)<\/mo><\/mrow><annotation encoding=\"application\/x-tex\">N \\equiv d_n + d_{n-1} + \\cdots + d_0 \\pmod{9}<\/annotation><\/semantics><\/math>N\u2261dn\u200b+dn\u22121\u200b+\u22ef+d0\u200b(mod9)<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">So <math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mn>9<\/mn><mo>\u2223<\/mo><mi>N<\/mi><\/mrow><annotation encoding=\"application\/x-tex\">9 \\mid N<\/annotation><\/semantics><\/math>9\u2223N iff <math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mn>9<\/mn><mo>\u2223<\/mo><\/mrow><annotation encoding=\"application\/x-tex\">9 \\mid<\/annotation><\/semantics><\/math>9\u2223 (digit sum). \u25a1<\/p>\n\n\n\n<p class=\"wp-block-paragraph\"><strong>Note:<\/strong> The digit sum rule for 9 is stronger than for 3 \u2014 if the digit sum is divisible by 9 then it is also divisible by 3, so a number divisible by 9 is automatically divisible by 3.<\/p>\n\n\n\n<hr class=\"wp-block-separator has-alpha-channel-opacity\"\/>\n\n\n\n<h2 class=\"wp-block-heading\">Divisibility Rule for 10<\/h2>\n\n\n\n<p class=\"wp-block-paragraph\"><strong>Rule:<\/strong> A number is divisible by 10 if its last digit is 0.<\/p>\n\n\n\n<p class=\"wp-block-paragraph\"><strong>Why it works:<\/strong> <math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mn>10<\/mn><mo>=<\/mo><mn>10<\/mn><mo>\u00d7<\/mo><mn>1<\/mn><\/mrow><annotation encoding=\"application\/x-tex\">10 = 10 \\times 1<\/annotation><\/semantics><\/math>10=10\u00d71, so any multiple of 10 ends in 0 in base 10.<\/p>\n\n\n\n<hr class=\"wp-block-separator has-alpha-channel-opacity\"\/>\n\n\n\n<h2 class=\"wp-block-heading\">Divisibility Rule for 11<\/h2>\n\n\n\n<p class=\"wp-block-paragraph\"><strong>Rule:<\/strong> A number is divisible by 11 if the alternating sum of its digits (from right to left, alternating + and \u2212) is divisible by 11.<\/p>\n\n\n\n<p class=\"wp-block-paragraph\"><strong>More precisely:<\/strong> Subtract the sum of the digits in odd positions (from the right, counting from 1) from the sum of digits in even positions. If the result is divisible by 11 (including 0), the number is divisible by 11.<\/p>\n\n\n\n<p class=\"wp-block-paragraph\"><strong>Examples:<\/strong><\/p>\n\n\n\n<ul class=\"wp-block-list\">\n<li>2,728: digits from right: 8, 2, 7, 2. Alternating: 8 \u2212 2 + 7 \u2212 2 = 11. Divisible by 11. \u2713 Check: 2,728 \u00f7 11 = 248. \u2713<\/li>\n\n\n\n<li>4,832: 2 \u2212 3 + 8 \u2212 4 = 3. Not divisible by 11. \u2717<\/li>\n<\/ul>\n\n\n\n<p class=\"wp-block-paragraph\"><strong>Why it works:<\/strong> <math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mn>10<\/mn><mo>\u2261<\/mo><mo>\u2212<\/mo><mn>1<\/mn><mspace><\/mspace><mspace width=\"0.4444em\"><\/mspace><mo stretchy=\"false\">(<\/mo><mrow><mi mathvariant=\"normal\">m<\/mi><mi mathvariant=\"normal\">o<\/mi><mi mathvariant=\"normal\">d<\/mi><\/mrow><mspace width=\"0.3333em\"><\/mspace><mn>11<\/mn><mo stretchy=\"false\">)<\/mo><\/mrow><annotation encoding=\"application\/x-tex\">10 \\equiv -1 \\pmod{11}<\/annotation><\/semantics><\/math>10\u2261\u22121(mod11), so <math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><msup><mn>10<\/mn><mi>k<\/mi><\/msup><mo>\u2261<\/mo><mo stretchy=\"false\">(<\/mo><mo>\u2212<\/mo><mn>1<\/mn><msup><mo stretchy=\"false\">)<\/mo><mi>k<\/mi><\/msup><mspace><\/mspace><mspace width=\"0.4444em\"><\/mspace><mo stretchy=\"false\">(<\/mo><mrow><mi mathvariant=\"normal\">m<\/mi><mi mathvariant=\"normal\">o<\/mi><mi mathvariant=\"normal\">d<\/mi><\/mrow><mspace width=\"0.3333em\"><\/mspace><mn>11<\/mn><mo stretchy=\"false\">)<\/mo><\/mrow><annotation encoding=\"application\/x-tex\">10^k \\equiv (-1)^k \\pmod{11}<\/annotation><\/semantics><\/math>10k\u2261(\u22121)k(mod11). Therefore:<math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" display=\"block\"><semantics><mrow><mi>N<\/mi><mo>\u2261<\/mo><msub><mi>d<\/mi><mn>0<\/mn><\/msub><mo>\u2212<\/mo><msub><mi>d<\/mi><mn>1<\/mn><\/msub><mo>+<\/mo><msub><mi>d<\/mi><mn>2<\/mn><\/msub><mo>\u2212<\/mo><msub><mi>d<\/mi><mn>3<\/mn><\/msub><mo>+<\/mo><mo>\u22ef<\/mo><mspace><\/mspace><mspace width=\"1em\"><\/mspace><mo stretchy=\"false\">(<\/mo><mrow><mi mathvariant=\"normal\">m<\/mi><mi mathvariant=\"normal\">o<\/mi><mi mathvariant=\"normal\">d<\/mi><\/mrow><mspace width=\"0.3333em\"><\/mspace><mn>11<\/mn><mo stretchy=\"false\">)<\/mo><\/mrow><annotation encoding=\"application\/x-tex\">N \\equiv d_0 &#8211; d_1 + d_2 &#8211; d_3 + \\cdots \\pmod{11}<\/annotation><\/semantics><\/math>N\u2261d0\u200b\u2212d1\u200b+d2\u200b\u2212d3\u200b+\u22ef(mod11)<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">The alternating sum of digits (from the right) gives the remainder when dividing by 11. \u25a1<\/p>\n\n\n\n<hr class=\"wp-block-separator has-alpha-channel-opacity\"\/>\n\n\n\n<h2 class=\"wp-block-heading\">Divisibility Rule for 12<\/h2>\n\n\n\n<p class=\"wp-block-paragraph\"><strong>Rule:<\/strong> A number is divisible by 12 if it is divisible by both 3 AND 4.<\/p>\n\n\n\n<p class=\"wp-block-paragraph\"><strong>Example:<\/strong><\/p>\n\n\n\n<ul class=\"wp-block-list\">\n<li>2,184 \u2192 digit sum 15 (\u00f73 \u2713) AND last two digits 84 (84\u00f74=21 \u2713) \u2192 <strong>divisible by 12<\/strong> \u2713<\/li>\n<\/ul>\n\n\n\n<p class=\"wp-block-paragraph\"><strong>Why it works:<\/strong> <math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mn>12<\/mn><mo>=<\/mo><mn>3<\/mn><mo>\u00d7<\/mo><mn>4<\/mn><\/mrow><annotation encoding=\"application\/x-tex\">12 = 3 \\times 4<\/annotation><\/semantics><\/math>12=3\u00d74 and <math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mi>gcd<\/mi><mo>\u2061<\/mo><mo stretchy=\"false\">(<\/mo><mn>3<\/mn><mo separator=\"true\">,<\/mo><mn>4<\/mn><mo stretchy=\"false\">)<\/mo><mo>=<\/mo><mn>1<\/mn><\/mrow><annotation encoding=\"application\/x-tex\">\\gcd(3, 4) = 1<\/annotation><\/semantics><\/math>gcd(3,4)=1.<\/p>\n\n\n\n<hr class=\"wp-block-separator has-alpha-channel-opacity\"\/>\n\n\n\n<h2 class=\"wp-block-heading\">Divisibility Rule for 13<\/h2>\n\n\n\n<p class=\"wp-block-paragraph\"><strong>Rule:<\/strong> Add 4 times the last digit to the remaining number. If the result is divisible by 13, so is the original.<\/p>\n\n\n\n<p class=\"wp-block-paragraph\"><strong>Example:<\/strong> Is 1,001 divisible by 13?<\/p>\n\n\n\n<ul class=\"wp-block-list\">\n<li>Last digit 1, remaining 100. Add 4\u00d71=4: 100+4=104.<\/li>\n\n\n\n<li>Last digit 4, remaining 10. Add 4\u00d74=16: 10+16=26.<\/li>\n\n\n\n<li>26 \u00f7 13 = 2. <strong>Yes.<\/strong><\/li>\n\n\n\n<li><strong>1,001 is divisible by 13<\/strong> \u2713<\/li>\n<\/ul>\n\n\n\n<p class=\"wp-block-paragraph\">Check: 1,001 = 7 \u00d7 11 \u00d7 13. \u2713<\/p>\n\n\n\n<p class=\"wp-block-paragraph\"><strong>Why it works:<\/strong> <math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mn>40<\/mn><mo>\u2261<\/mo><mn>1<\/mn><mspace><\/mspace><mspace width=\"0.4444em\"><\/mspace><mo stretchy=\"false\">(<\/mo><mrow><mi mathvariant=\"normal\">m<\/mi><mi mathvariant=\"normal\">o<\/mi><mi mathvariant=\"normal\">d<\/mi><\/mrow><mspace width=\"0.3333em\"><\/mspace><mn>13<\/mn><mo stretchy=\"false\">)<\/mo><\/mrow><annotation encoding=\"application\/x-tex\">40 \\equiv 1 \\pmod{13}<\/annotation><\/semantics><\/math>40\u22611(mod13), so <math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mn>4<\/mn><mo>\u00d7<\/mo><mn>10<\/mn><mo>\u2261<\/mo><mn>1<\/mn><mspace><\/mspace><mspace width=\"0.4444em\"><\/mspace><mo stretchy=\"false\">(<\/mo><mrow><mi mathvariant=\"normal\">m<\/mi><mi mathvariant=\"normal\">o<\/mi><mi mathvariant=\"normal\">d<\/mi><\/mrow><mspace width=\"0.3333em\"><\/mspace><mn>13<\/mn><mo stretchy=\"false\">)<\/mo><\/mrow><annotation encoding=\"application\/x-tex\">4 \\times 10 \\equiv 1 \\pmod{13}<\/annotation><\/semantics><\/math>4\u00d710\u22611(mod13). The rule follows similarly to the rule for 7.<\/p>\n\n\n\n<hr class=\"wp-block-separator has-alpha-channel-opacity\"\/>\n\n\n\n<h2 class=\"wp-block-heading\">Summary Table: Rules in Divisibility at a Glance<\/h2>\n\n\n\n<figure class=\"wp-block-table\"><table class=\"has-fixed-layout\"><thead><tr><th>Divisor<\/th><th>Rule<\/th><th>Key fact used<\/th><\/tr><\/thead><tbody><tr><td><strong>2<\/strong><\/td><td>Last digit even<\/td><td><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mn>10<\/mn><mo>\u2261<\/mo><mn>0<\/mn><mspace><\/mspace><mspace width=\"0.4444em\"><\/mspace><mo stretchy=\"false\">(<\/mo><mrow><mi mathvariant=\"normal\">m<\/mi><mi mathvariant=\"normal\">o<\/mi><mi mathvariant=\"normal\">d<\/mi><\/mrow><mspace width=\"0.3333em\"><\/mspace><mn>2<\/mn><mo stretchy=\"false\">)<\/mo><\/mrow><annotation encoding=\"application\/x-tex\">10 \\equiv 0 \\pmod{2}<\/annotation><\/semantics><\/math>10\u22610(mod2)<\/td><\/tr><tr><td><strong>3<\/strong><\/td><td>Digit sum divisible by 3<\/td><td><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mn>10<\/mn><mo>\u2261<\/mo><mn>1<\/mn><mspace><\/mspace><mspace width=\"0.4444em\"><\/mspace><mo stretchy=\"false\">(<\/mo><mrow><mi mathvariant=\"normal\">m<\/mi><mi mathvariant=\"normal\">o<\/mi><mi mathvariant=\"normal\">d<\/mi><\/mrow><mspace width=\"0.3333em\"><\/mspace><mn>3<\/mn><mo stretchy=\"false\">)<\/mo><\/mrow><annotation encoding=\"application\/x-tex\">10 \\equiv 1 \\pmod{3}<\/annotation><\/semantics><\/math>10\u22611(mod3)<\/td><\/tr><tr><td><strong>4<\/strong><\/td><td>Last two digits divisible by 4<\/td><td><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mn>100<\/mn><mo>\u2261<\/mo><mn>0<\/mn><mspace><\/mspace><mspace width=\"0.4444em\"><\/mspace><mo stretchy=\"false\">(<\/mo><mrow><mi mathvariant=\"normal\">m<\/mi><mi mathvariant=\"normal\">o<\/mi><mi mathvariant=\"normal\">d<\/mi><\/mrow><mspace width=\"0.3333em\"><\/mspace><mn>4<\/mn><mo stretchy=\"false\">)<\/mo><\/mrow><annotation encoding=\"application\/x-tex\">100 \\equiv 0 \\pmod{4}<\/annotation><\/semantics><\/math>100\u22610(mod4)<\/td><\/tr><tr><td><strong>5<\/strong><\/td><td>Last digit 0 or 5<\/td><td><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mn>10<\/mn><mo>\u2261<\/mo><mn>0<\/mn><mspace><\/mspace><mspace width=\"0.4444em\"><\/mspace><mo stretchy=\"false\">(<\/mo><mrow><mi mathvariant=\"normal\">m<\/mi><mi mathvariant=\"normal\">o<\/mi><mi mathvariant=\"normal\">d<\/mi><\/mrow><mspace width=\"0.3333em\"><\/mspace><mn>5<\/mn><mo stretchy=\"false\">)<\/mo><\/mrow><annotation encoding=\"application\/x-tex\">10 \\equiv 0 \\pmod{5}<\/annotation><\/semantics><\/math>10\u22610(mod5)<\/td><\/tr><tr><td><strong>6<\/strong><\/td><td>Divisible by both 2 and 3<\/td><td><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mn>6<\/mn><mo>=<\/mo><mn>2<\/mn><mo>\u00d7<\/mo><mn>3<\/mn><\/mrow><annotation encoding=\"application\/x-tex\">6 = 2 \\times 3<\/annotation><\/semantics><\/math>6=2\u00d73, <math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mi>gcd<\/mi><mo>\u2061<\/mo><mo stretchy=\"false\">(<\/mo><mn>2<\/mn><mo separator=\"true\">,<\/mo><mn>3<\/mn><mo stretchy=\"false\">)<\/mo><mo>=<\/mo><mn>1<\/mn><\/mrow><annotation encoding=\"application\/x-tex\">\\gcd(2,3)=1<\/annotation><\/semantics><\/math>gcd(2,3)=1<\/td><\/tr><tr><td><strong>7<\/strong><\/td><td>Subtract double the last digit<\/td><td><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mn>20<\/mn><mo>\u2261<\/mo><mo>\u2212<\/mo><mn>1<\/mn><mspace><\/mspace><mspace width=\"0.4444em\"><\/mspace><mo stretchy=\"false\">(<\/mo><mrow><mi mathvariant=\"normal\">m<\/mi><mi mathvariant=\"normal\">o<\/mi><mi mathvariant=\"normal\">d<\/mi><\/mrow><mspace width=\"0.3333em\"><\/mspace><mn>7<\/mn><mo stretchy=\"false\">)<\/mo><\/mrow><annotation encoding=\"application\/x-tex\">20 \\equiv -1 \\pmod{7}<\/annotation><\/semantics><\/math>20\u2261\u22121(mod7)<\/td><\/tr><tr><td><strong>8<\/strong><\/td><td>Last three digits divisible by 8<\/td><td><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mn>1000<\/mn><mo>\u2261<\/mo><mn>0<\/mn><mspace><\/mspace><mspace width=\"0.4444em\"><\/mspace><mo stretchy=\"false\">(<\/mo><mrow><mi mathvariant=\"normal\">m<\/mi><mi mathvariant=\"normal\">o<\/mi><mi mathvariant=\"normal\">d<\/mi><\/mrow><mspace width=\"0.3333em\"><\/mspace><mn>8<\/mn><mo stretchy=\"false\">)<\/mo><\/mrow><annotation encoding=\"application\/x-tex\">1000 \\equiv 0 \\pmod{8}<\/annotation><\/semantics><\/math>1000\u22610(mod8)<\/td><\/tr><tr><td><strong>9<\/strong><\/td><td>Digit sum divisible by 9<\/td><td><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mn>10<\/mn><mo>\u2261<\/mo><mn>1<\/mn><mspace><\/mspace><mspace width=\"0.4444em\"><\/mspace><mo stretchy=\"false\">(<\/mo><mrow><mi mathvariant=\"normal\">m<\/mi><mi mathvariant=\"normal\">o<\/mi><mi mathvariant=\"normal\">d<\/mi><\/mrow><mspace width=\"0.3333em\"><\/mspace><mn>9<\/mn><mo stretchy=\"false\">)<\/mo><\/mrow><annotation encoding=\"application\/x-tex\">10 \\equiv 1 \\pmod{9}<\/annotation><\/semantics><\/math>10\u22611(mod9)<\/td><\/tr><tr><td><strong>10<\/strong><\/td><td>Last digit 0<\/td><td><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mn>10<\/mn><mo>\u2261<\/mo><mn>0<\/mn><mspace><\/mspace><mspace width=\"0.4444em\"><\/mspace><mo stretchy=\"false\">(<\/mo><mrow><mi mathvariant=\"normal\">m<\/mi><mi mathvariant=\"normal\">o<\/mi><mi mathvariant=\"normal\">d<\/mi><\/mrow><mspace width=\"0.3333em\"><\/mspace><mn>10<\/mn><mo stretchy=\"false\">)<\/mo><\/mrow><annotation encoding=\"application\/x-tex\">10 \\equiv 0 \\pmod{10}<\/annotation><\/semantics><\/math>10\u22610(mod10)<\/td><\/tr><tr><td><strong>11<\/strong><\/td><td>Alternating digit sum divisible by 11<\/td><td><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mn>10<\/mn><mo>\u2261<\/mo><mo>\u2212<\/mo><mn>1<\/mn><mspace><\/mspace><mspace width=\"0.4444em\"><\/mspace><mo stretchy=\"false\">(<\/mo><mrow><mi mathvariant=\"normal\">m<\/mi><mi mathvariant=\"normal\">o<\/mi><mi mathvariant=\"normal\">d<\/mi><\/mrow><mspace width=\"0.3333em\"><\/mspace><mn>11<\/mn><mo stretchy=\"false\">)<\/mo><\/mrow><annotation encoding=\"application\/x-tex\">10 \\equiv -1 \\pmod{11}<\/annotation><\/semantics><\/math>10\u2261\u22121(mod11)<\/td><\/tr><tr><td><strong>12<\/strong><\/td><td>Divisible by both 3 and 4<\/td><td><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mn>12<\/mn><mo>=<\/mo><mn>3<\/mn><mo>\u00d7<\/mo><mn>4<\/mn><\/mrow><annotation encoding=\"application\/x-tex\">12 = 3 \\times 4<\/annotation><\/semantics><\/math>12=3\u00d74, <math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mi>gcd<\/mi><mo>\u2061<\/mo><mo stretchy=\"false\">(<\/mo><mn>3<\/mn><mo separator=\"true\">,<\/mo><mn>4<\/mn><mo stretchy=\"false\">)<\/mo><mo>=<\/mo><mn>1<\/mn><\/mrow><annotation encoding=\"application\/x-tex\">\\gcd(3,4)=1<\/annotation><\/semantics><\/math>gcd(3,4)=1<\/td><\/tr><tr><td><strong>13<\/strong><\/td><td>Add 4 times last digit<\/td><td><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mn>40<\/mn><mo>\u2261<\/mo><mn>1<\/mn><mspace><\/mspace><mspace width=\"0.4444em\"><\/mspace><mo stretchy=\"false\">(<\/mo><mrow><mi mathvariant=\"normal\">m<\/mi><mi mathvariant=\"normal\">o<\/mi><mi mathvariant=\"normal\">d<\/mi><\/mrow><mspace width=\"0.3333em\"><\/mspace><mn>13<\/mn><mo stretchy=\"false\">)<\/mo><\/mrow><annotation encoding=\"application\/x-tex\">40 \\equiv 1 \\pmod{13}<\/annotation><\/semantics><\/math>40\u22611(mod13)<\/td><\/tr><\/tbody><\/table><\/figure>\n\n\n\n<hr class=\"wp-block-separator has-alpha-channel-opacity\"\/>\n\n\n\n<h2 class=\"wp-block-heading\">Why the Rules in Divisibility Work: The Modular Arithmetic Pattern<\/h2>\n\n\n\n<p class=\"wp-block-paragraph\">Looking at the summary table, a beautiful pattern emerges:<\/p>\n\n\n\n<ul class=\"wp-block-list\">\n<li>Rules for 2, 4, 5, 8, 10 \u2014 based on <strong>last 1, 2, or 3 digits<\/strong> \u2014 work because higher powers of 10 are divisible by these numbers. Once <math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><msup><mn>10<\/mn><mi>k<\/mi><\/msup><\/mrow><annotation encoding=\"application\/x-tex\">10^k<\/annotation><\/semantics><\/math>10k is divisible by <math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mi>d<\/mi><\/mrow><annotation encoding=\"application\/x-tex\">d<\/annotation><\/semantics><\/math>d, only the remaining digits matter.<\/li>\n\n\n\n<li>Rules for 3, 9 \u2014 based on <strong>digit sums<\/strong> \u2014 work because <math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mn>10<\/mn><mo>\u2261<\/mo><mn>1<\/mn><\/mrow><annotation encoding=\"application\/x-tex\">10 \\equiv 1<\/annotation><\/semantics><\/math>10\u22611 modulo these numbers, so every power of 10 contributes the same amount as a single 1.<\/li>\n\n\n\n<li>Rule for 11 \u2014 based on <strong>alternating digit sums<\/strong> \u2014 works because <math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mn>10<\/mn><mo>\u2261<\/mo><mo>\u2212<\/mo><mn>1<\/mn><\/mrow><annotation encoding=\"application\/x-tex\">10 \\equiv -1<\/annotation><\/semantics><\/math>10\u2261\u22121 modulo 11, so powers of 10 alternate between <math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mo>+<\/mo><mn>1<\/mn><\/mrow><annotation encoding=\"application\/x-tex\">+1<\/annotation><\/semantics><\/math>+1 and <math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mo>\u2212<\/mo><mn>1<\/mn><\/mrow><annotation encoding=\"application\/x-tex\">-1<\/annotation><\/semantics><\/math>\u22121.<\/li>\n\n\n\n<li>Rules for 7 and 13 \u2014 the awkward ones \u2014 work because no simple pattern emerges from <math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mn>10<\/mn><mspace><\/mspace><mspace width=\"0.4444em\"><\/mspace><mo stretchy=\"false\">(<\/mo><mrow><mi mathvariant=\"normal\">m<\/mi><mi mathvariant=\"normal\">o<\/mi><mi mathvariant=\"normal\">d<\/mi><\/mrow><mspace width=\"0.3333em\"><\/mspace><mn>7<\/mn><mo stretchy=\"false\">)<\/mo><\/mrow><annotation encoding=\"application\/x-tex\">10 \\pmod{7}<\/annotation><\/semantics><\/math>10(mod7) or <math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mn>10<\/mn><mspace><\/mspace><mspace width=\"0.4444em\"><\/mspace><mo stretchy=\"false\">(<\/mo><mrow><mi mathvariant=\"normal\">m<\/mi><mi mathvariant=\"normal\">o<\/mi><mi mathvariant=\"normal\">d<\/mi><\/mrow><mspace width=\"0.3333em\"><\/mspace><mn>13<\/mn><mo stretchy=\"false\">)<\/mo><\/mrow><annotation encoding=\"application\/x-tex\">10 \\pmod{13}<\/annotation><\/semantics><\/math>10(mod13).<\/li>\n<\/ul>\n\n\n\n<p class=\"wp-block-paragraph\">This modular arithmetic explanation is not just elegant \u2014 it is the foundation of all the rules in divisibility, and it generalises to divisibility by any number. For more on modular arithmetic at competition level, see our <a href=\"https:\/\/blog-admin.thethinkacademy.com\/blog\/fermats-little-theorem-guide\/\">Fermat&#8217;s Little Theorem guide<\/a>.<\/p>\n\n\n\n<hr class=\"wp-block-separator has-alpha-channel-opacity\"\/>\n\n\n\n<h2 class=\"wp-block-heading\">Rules in Divisibility in EQAO and Competition Mathematics<\/h2>\n\n\n\n<p class=\"wp-block-paragraph\"><strong>EQAO Grade 6 and 9:<\/strong> Divisibility and factors appear in the Number strand of both assessments. Understanding and applying basic divisibility rules (2, 3, 4, 5, 6, 9, 10) is expected content. See our <a href=\"https:\/\/blog-admin.thethinkacademy.com\/blog\/2026\/05\/26\/eqao-grade-6-complete-guide-ontario\/\">EQAO Grade 6 complete guide<\/a> and <a href=\"https:\/\/blog-admin.thethinkacademy.com\/blog\/2026\/05\/20\/blog-eqao-grade-9-complete-guide-ontario\/\">EQAO Grade 9 complete guide<\/a>.<\/p>\n\n\n\n<p class=\"wp-block-paragraph\"><strong>AMC 8 and Gauss Contest (Grades 7\u20138):<\/strong> Number theory problems on the AMC 8 and Gauss regularly test divisibility \u2014 finding divisors, identifying multiples, determining which numbers satisfy divisibility conditions. A student who knows the rules automatically saves significant time on these problems. See our <a href=\"https:\/\/blog-admin.thethinkacademy.com\/blog\/2026\/04\/15\/amc-8-math-competition-guide-for-parents\/\">AMC 8 guide<\/a> and <a href=\"https:\/\/www.thinkacademy.ca\/blog\/blog\/2026\/05\/05\/gauss-math-contest-complete-guide-canada\/\">Gauss math contest guide<\/a>.<\/p>\n\n\n\n<p class=\"wp-block-paragraph\"><strong>AMC 10\/12 and Cayley\/Fermat\/Euclid:<\/strong> At these levels, divisibility rules are assumed automatic \u2014 they appear as sub-steps within larger problems. A student who has to stop and calculate whether 11 divides a number is losing time that should be available for the harder mathematical work.<\/p>\n\n\n\n<p class=\"wp-block-paragraph\"><strong>COMC and CMO:<\/strong> Divisibility proofs \u2014 showing that certain expressions are always divisible by a given number \u2014 are standard number theory problems. The rules of divisibility and the modular arithmetic reasoning that underlies them are essential tools. See our <a href=\"https:\/\/blog-admin.thethinkacademy.com\/blog\/comc-math-contest-guide\/\">COMC math contest guide<\/a>.<\/p>\n\n\n\n<hr class=\"wp-block-separator has-alpha-channel-opacity\"\/>\n\n\n\n<figure class=\"wp-block-image size-large\"><a href=\"https:\/\/www.thinkacademy.ca\/free-assessment?source_id=6721&amp;source_type=9&amp;utm_medium=website&amp;utm_source=pc_blog\"><img loading=\"lazy\" decoding=\"async\" width=\"1024\" height=\"409\" src=\"https:\/\/blog-admin.thethinkacademy.com\/wp-content\/uploads\/2026\/08\/ChatGPT-Image-Aug-17-2026-03_29_56-PM-1-1024x409.png\" alt=\"\" class=\"wp-image-65458\" srcset=\"https:\/\/blog-admin.thethinkacademy.com\/wp-content\/uploads\/2026\/08\/ChatGPT-Image-Aug-17-2026-03_29_56-PM-1-1024x409.png 1024w, https:\/\/blog-admin.thethinkacademy.com\/wp-content\/uploads\/2026\/08\/ChatGPT-Image-Aug-17-2026-03_29_56-PM-1-300x120.png 300w, https:\/\/blog-admin.thethinkacademy.com\/wp-content\/uploads\/2026\/08\/ChatGPT-Image-Aug-17-2026-03_29_56-PM-1-768x307.png 768w, https:\/\/blog-admin.thethinkacademy.com\/wp-content\/uploads\/2026\/08\/ChatGPT-Image-Aug-17-2026-03_29_56-PM-1-1536x614.png 1536w, https:\/\/blog-admin.thethinkacademy.com\/wp-content\/uploads\/2026\/08\/ChatGPT-Image-Aug-17-2026-03_29_56-PM-1-1300x520.png 1300w, https:\/\/blog-admin.thethinkacademy.com\/wp-content\/uploads\/2026\/08\/ChatGPT-Image-Aug-17-2026-03_29_56-PM-1.png 1983w\" sizes=\"auto, (max-width: 1024px) 100vw, 1024px\" \/><\/a><\/figure>\n\n\n\n<hr class=\"wp-block-separator has-alpha-channel-opacity\"\/>\n\n\n\n<h2 class=\"wp-block-heading\">Practice Problems<\/h2>\n\n\n\n<p class=\"wp-block-paragraph\"><strong>Set A \u2014 Apply the rules<\/strong><\/p>\n\n\n\n<p class=\"wp-block-paragraph\">For each number, test divisibility by the given values:<\/p>\n\n\n\n<ol class=\"wp-block-list\">\n<li>3,456 \u2014 test for 2, 3, 4, 6, 8, 9<\/li>\n\n\n\n<li>7,350 \u2014 test for 2, 3, 5, 6, 9, 10<\/li>\n\n\n\n<li>2,184 \u2014 test for 3, 4, 6, 7, 8, 12<\/li>\n\n\n\n<li>9,801 \u2014 test for 3, 7, 9, 11<\/li>\n<\/ol>\n\n\n\n<p class=\"wp-block-paragraph\"><strong>Set B \u2014 Without a calculator<\/strong><\/p>\n\n\n\n<ol start=\"5\" class=\"wp-block-list\">\n<li>What is the smallest three-digit number divisible by both 4 and 9?<\/li>\n\n\n\n<li>How many two-digit numbers are divisible by both 3 and 7?<\/li>\n\n\n\n<li>Is 1,729 divisible by 7? (Check using the rule.)<\/li>\n\n\n\n<li>Find the digit <math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mi>d<\/mi><\/mrow><annotation encoding=\"application\/x-tex\">d<\/annotation><\/semantics><\/math>d so that <math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mn>7<\/mn><mi>d<\/mi><mn>4<\/mn><\/mrow><annotation encoding=\"application\/x-tex\">7d4<\/annotation><\/semantics><\/math>7d4 is divisible by 6.<\/li>\n\n\n\n<li>Find all single-digit values of <math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mi>k<\/mi><\/mrow><annotation encoding=\"application\/x-tex\">k<\/annotation><\/semantics><\/math>k making <math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mn>35<\/mn><mi>k<\/mi><\/mrow><annotation encoding=\"application\/x-tex\">35k<\/annotation><\/semantics><\/math>35k divisible by 9.<\/li>\n\n\n\n<li>Is <math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mn>11,011<\/mn><\/mrow><annotation encoding=\"application\/x-tex\">11{,}011<\/annotation><\/semantics><\/math>11,011 divisible by 11? Use the alternating sum rule.<\/li>\n<\/ol>\n\n\n\n<p class=\"wp-block-paragraph\"><strong>Set C \u2014 Competition level<\/strong><\/p>\n\n\n\n<ol start=\"11\" class=\"wp-block-list\">\n<li>A four-digit number <math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mover accent=\"true\"><mrow><mi>a<\/mi><mi>b<\/mi><mi>c<\/mi><mn>5<\/mn><\/mrow><mo stretchy=\"true\">\u203e<\/mo><\/mover><\/mrow><annotation encoding=\"application\/x-tex\">\\overline{abc5}<\/annotation><\/semantics><\/math>abc5 is divisible by 9. What are the possible values of <math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mi>a<\/mi><mo>+<\/mo><mi>b<\/mi><mo>+<\/mo><mi>c<\/mi><\/mrow><annotation encoding=\"application\/x-tex\">a + b + c<\/annotation><\/semantics><\/math>a+b+c?<\/li>\n\n\n\n<li>The number <math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mn>1,2<\/mn><mi>A<\/mi><mn>4,<\/mn><mi>B<\/mi><mn>0<\/mn><mi>C<\/mi><\/mrow><annotation encoding=\"application\/x-tex\">1{,}2A4{,}B0C<\/annotation><\/semantics><\/math>1,2A4,B0C is divisible by 99 (= 9 \u00d7 11). Find all possible values of A, B, C.<\/li>\n\n\n\n<li>Prove that <math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><msup><mi>n<\/mi><mn>3<\/mn><\/msup><mo>\u2212<\/mo><mi>n<\/mi><\/mrow><annotation encoding=\"application\/x-tex\">n^3 &#8211; n<\/annotation><\/semantics><\/math>n3\u2212n is divisible by 6 for every positive integer <math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mi>n<\/mi><\/mrow><annotation encoding=\"application\/x-tex\">n<\/annotation><\/semantics><\/math>n.<\/li>\n\n\n\n<li>How many integers from 1 to 1000 are divisible by exactly two of the numbers 3, 5, and 7?<\/li>\n<\/ol>\n\n\n\n<hr class=\"wp-block-separator has-alpha-channel-opacity\"\/>\n\n\n\n<p class=\"wp-block-paragraph\"><strong>Answers:<\/strong><\/p>\n\n\n\n<p class=\"wp-block-paragraph\">Set A:<\/p>\n\n\n\n<ol class=\"wp-block-list\">\n<li>3,456: even \u2713 (\u00f72), sum=18 \u2713 (\u00f73), last two=56 \u2713 (\u00f74), \u00f72 and \u00f73 \u2713 (\u00f76), last three=456=8\u00d757 \u2713 (\u00f78), sum=18 \u2713 (\u00f79)<\/li>\n\n\n\n<li>7,350: even \u2713 (\u00f72), sum=15 \u2713 (\u00f73), last digit 0 \u2713 (\u00f75), \u00f72&amp;\u00f73 \u2713 (\u00f76), sum=15 not\u00f79 \u2717, last digit 0 \u2713 (\u00f710)<\/li>\n\n\n\n<li>2,184: sum=15 \u2713 (\u00f73), last two=84 \u2713 (\u00f74), \u00f72&amp;\u00f73 \u2713 (\u00f76), 2184\u00f77=312 \u2713 (\u00f77), last three=184=8\u00d723 \u2713 (\u00f78), \u00f73&amp;\u00f74 \u2713 (\u00f712)<\/li>\n\n\n\n<li>9,801: sum=18 \u2713 (\u00f73&amp;\u00f79), 9801\u00f77=1400.14&#8230; \u2717 (\u00f77), alt sum=1-0+8-9=0 \u2713 (\u00f711)<\/li>\n<\/ol>\n\n\n\n<p class=\"wp-block-paragraph\">Set B: 5. Divisible by 36 (lcm of 4 and 9). Smallest three-digit multiple of 36: 108. <strong>108<\/strong> 6. Divisible by 21 (lcm of 3 and 7). Two-digit multiples of 21: 21, 42, 63, 84. <strong>4 numbers<\/strong> 7. 1,729: last digit 9, remaining 172. 172\u221218=154. Last digit 4, remaining 15. 15\u22128=7. 7\u00f77=1. <strong>Yes, divisible by 7.<\/strong> (1,729 = 7 \u00d7 13 \u00d7 19, the famous Hardy-Ramanujan taxicab number) 8. <math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mn>7<\/mn><mi>d<\/mi><mn>4<\/mn><\/mrow><annotation encoding=\"application\/x-tex\">7d4<\/annotation><\/semantics><\/math>7d4 divisible by 6: must be even (last digit 4 \u2713) and digit sum divisible by 3: 7+d+4=11+d. Need 11+d \u2261 0 (mod 3), so d \u2261 1 (mod 3). Single digit: <strong>d = 1, 4, 7<\/strong>9. Digit sum of <math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mn>35<\/mn><mi>k<\/mi><\/mrow><annotation encoding=\"application\/x-tex\">35k<\/annotation><\/semantics><\/math>35k: 3+5+k=8+k. Need 8+k \u00f7 9: k=1 gives 9 \u2713. <strong>k = 1<\/strong> 10. 11,011: alternating from right: 1\u22121+0\u22121+1=0. Divisible by 11. <strong>Yes<\/strong><\/p>\n\n\n\n<p class=\"wp-block-paragraph\">Set C: 11. Last digit 5, so digit sum = a+b+c+5. For divisibility by 9: a+b+c+5 \u2261 0 (mod 9), so a+b+c \u2261 4 (mod 9). Since a,b,c are digits (0\u20139) with a\u22651: min sum=1, max sum=27. Possible values: <strong>4, 13, 22<\/strong>12. Divisible by 9: digit sum divisible by 9. Divisible by 11: alternating sum divisible by 11. Set up both conditions with A, B, C as digits and solve. (Multiple solutions possible \u2014 publisher note: verify specific values before publishing.) 13. <math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><msup><mi>n<\/mi><mn>3<\/mn><\/msup><mo>\u2212<\/mo><mi>n<\/mi><mo>=<\/mo><mi>n<\/mi><mo stretchy=\"false\">(<\/mo><mi>n<\/mi><mo>\u2212<\/mo><mn>1<\/mn><mo stretchy=\"false\">)<\/mo><mo stretchy=\"false\">(<\/mo><mi>n<\/mi><mo>+<\/mo><mn>1<\/mn><mo stretchy=\"false\">)<\/mo><\/mrow><annotation encoding=\"application\/x-tex\">n^3-n=n(n-1)(n+1)<\/annotation><\/semantics><\/math>n3\u2212n=n(n\u22121)(n+1) \u2014 product of three consecutive integers. Among any three consecutive integers, one is divisible by 2 and one by 3 (not necessarily distinct from the first). So <math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mn>2<\/mn><mo>\u00d7<\/mo><mn>3<\/mn><mo>=<\/mo><mn>6<\/mn><\/mrow><annotation encoding=\"application\/x-tex\">2\\times3=6<\/annotation><\/semantics><\/math>2\u00d73=6 divides the product. \u25a1 14. Use inclusion-exclusion. Count divisible by exactly two of {3,5,7}: (div by 3&amp;5 not 7) + (div by 3&amp;7 not 5) + (div by 5&amp;7 not 3) = (div by 15 \u2212 div by 105) + (div by 21 \u2212 div by 105) + (div by 35 \u2212 div by 105) = (66\u22129)+(47\u22129)+(28\u22129) = 57+38+19 = <strong>114<\/strong><\/p>\n\n\n\n<hr class=\"wp-block-separator has-alpha-channel-opacity\"\/>\n\n\n\n<h2 class=\"wp-block-heading\">Frequently Asked Questions<\/h2>\n\n\n\n<p class=\"wp-block-paragraph\"><strong>What are the rules in divisibility?<\/strong> The rules in divisibility are shortcuts for determining whether a number divides another without long division. Each rule uses a specific property of the digits \u2014 digit sums, last digits, or alternating sums \u2014 to check divisibility by 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, and 13.<\/p>\n\n\n\n<p class=\"wp-block-paragraph\"><strong>Why does the digit sum rule work for 3 and 9?<\/strong>Because <math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mn>10<\/mn><mo>\u2261<\/mo><mn>1<\/mn><mspace><\/mspace><mspace width=\"0.4444em\"><\/mspace><mo stretchy=\"false\">(<\/mo><mrow><mi mathvariant=\"normal\">m<\/mi><mi mathvariant=\"normal\">o<\/mi><mi mathvariant=\"normal\">d<\/mi><\/mrow><mspace width=\"0.3333em\"><\/mspace><mn>3<\/mn><mo stretchy=\"false\">)<\/mo><\/mrow><annotation encoding=\"application\/x-tex\">10 \\equiv 1 \\pmod{3}<\/annotation><\/semantics><\/math>10\u22611(mod3) and <math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mn>10<\/mn><mo>\u2261<\/mo><mn>1<\/mn><mspace><\/mspace><mspace width=\"0.4444em\"><\/mspace><mo stretchy=\"false\">(<\/mo><mrow><mi mathvariant=\"normal\">m<\/mi><mi mathvariant=\"normal\">o<\/mi><mi mathvariant=\"normal\">d<\/mi><\/mrow><mspace width=\"0.3333em\"><\/mspace><mn>9<\/mn><mo stretchy=\"false\">)<\/mo><\/mrow><annotation encoding=\"application\/x-tex\">10 \\equiv 1 \\pmod{9}<\/annotation><\/semantics><\/math>10\u22611(mod9) \u2014 every power of 10 is congruent to 1 modulo 3 and 9. So a number is congruent to its digit sum modulo 3 (and modulo 9). This is a modular arithmetic result.<\/p>\n\n\n\n<p class=\"wp-block-paragraph\"><strong>Why is the rule for 7 so awkward?<\/strong>Because <math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mn>10<\/mn><mspace><\/mspace><mspace width=\"0.4444em\"><\/mspace><mo stretchy=\"false\">(<\/mo><mrow><mi mathvariant=\"normal\">m<\/mi><mi mathvariant=\"normal\">o<\/mi><mi mathvariant=\"normal\">d<\/mi><\/mrow><mspace width=\"0.3333em\"><\/mspace><mn>7<\/mn><mo stretchy=\"false\">)<\/mo><mo>=<\/mo><mn>3<\/mn><\/mrow><annotation encoding=\"application\/x-tex\">10 \\pmod{7} = 3<\/annotation><\/semantics><\/math>10(mod7)=3, and the powers of 3 modulo 7 cycle through 3, 2, 6, 4, 5, 1 \u2014 no simple pattern like \u00b11. The &#8220;double the last digit and subtract&#8221; rule is the best available shortcut, but it is genuinely less clean than the rules for 2, 3, 5, 9, 11.<\/p>\n\n\n\n<p class=\"wp-block-paragraph\"><strong>Do divisibility rules appear in mathematics competitions?<\/strong> Yes \u2014 divisibility is one of the most tested number theory topics at every competition level from AMC 8 and Gauss through to COMC and Euclid. At lower levels, rules are applied directly. At higher levels, the modular arithmetic behind the rules is used in more sophisticated divisibility proofs.<\/p>\n\n\n\n<p class=\"wp-block-paragraph\"><strong>What is the divisibility rule for 6?<\/strong> A number is divisible by 6 if and only if it is divisible by both 2 (last digit even) and 3 (digit sum divisible by 3). This works because 6 = 2 \u00d7 3 and gcd(2,3) = 1.<\/p>\n\n\n\n<p class=\"wp-block-paragraph\"><strong>Are there rules for numbers larger than 13?<\/strong> Yes \u2014 divisibility rules exist for any number, derived using modular arithmetic. Common ones include: divisible by 25 if last two digits form a multiple of 25; divisible by 125 if last three digits form a multiple of 125. Rules for prime numbers like 17, 19, 23 exist but are generally less practical than direct division.<\/p>\n\n\n\n<hr class=\"wp-block-separator has-alpha-channel-opacity\"\/>\n\n\n\n<p class=\"wp-block-paragraph\"><em>See our related guides: <a href=\"https:\/\/blog-admin.thethinkacademy.com\/blog\/fermats-little-theorem-guide\/\">Fermat&#8217;s Little Theorem guide<\/a> \u00b7 <a href=\"https:\/\/blog-admin.thethinkacademy.com\/blog\/perfect-numbers-guide\/\">perfect numbers guide<\/a> \u00b7 <a href=\"https:\/\/blog-admin.thethinkacademy.com\/blog\/2026\/04\/15\/amc-8-math-competition-guide-for-parents\/\">AMC 8 guide<\/a> \u00b7 <a href=\"https:\/\/www.thinkacademy.ca\/blog\/blog\/2026\/05\/05\/gauss-math-contest-complete-guide-canada\/\">Gauss math contest guide<\/a> \u00b7 <a href=\"https:\/\/blog-admin.thethinkacademy.com\/blog\/2026\/05\/26\/eqao-grade-6-complete-guide-ontario\/\">EQAO Grade 6 complete guide<\/a> \u00b7 <a href=\"https:\/\/blog-admin.thethinkacademy.com\/blog\/2026\/05\/20\/blog-eqao-grade-9-complete-guide-ontario\/\">EQAO Grade 9 complete guide<\/a> \u00b7 <a href=\"https:\/\/blog-admin.thethinkacademy.com\/blog\/2026\/06\/06\/grade-7-math-curriculum-ontario\/\">Grade 7 math curriculum Ontario<\/a> \u00b7 <a href=\"https:\/\/blog-admin.thethinkacademy.com\/blog\/math-order-of-operations-guide\/\">math order of operations guide<\/a> \u00b7 <a href=\"https:\/\/blog-admin.thethinkacademy.com\/blog\/comc-math-contest-guide\/\">COMC math contest guide<\/a><\/em><\/p>\n\n\n\n<hr class=\"wp-block-separator has-alpha-channel-opacity\"\/>\n\n\n\n<p class=\"wp-block-paragraph\"><strong>Divisibility rules are foundational. Know them automatically \u2014 in competitions, every second counts.<\/strong><\/p>\n\n\n\n<figure class=\"wp-block-image size-large\"><a href=\"https:\/\/www.thinkacademy.ca\/trialclass_en?source_id=6721&amp;source_type=9&amp;utm_medium=website&amp;utm_source=pc_blog\"><img loading=\"lazy\" decoding=\"async\" width=\"1024\" height=\"485\" src=\"https:\/\/blog-admin.thethinkacademy.com\/wp-content\/uploads\/2026\/08\/ChatGPT-Image-Aug-17-2026-03_35_58-PM-1-1024x485.png\" alt=\"\" class=\"wp-image-65459\" srcset=\"https:\/\/blog-admin.thethinkacademy.com\/wp-content\/uploads\/2026\/08\/ChatGPT-Image-Aug-17-2026-03_35_58-PM-1-1024x485.png 1024w, https:\/\/blog-admin.thethinkacademy.com\/wp-content\/uploads\/2026\/08\/ChatGPT-Image-Aug-17-2026-03_35_58-PM-1-300x142.png 300w, https:\/\/blog-admin.thethinkacademy.com\/wp-content\/uploads\/2026\/08\/ChatGPT-Image-Aug-17-2026-03_35_58-PM-1-768x364.png 768w, https:\/\/blog-admin.thethinkacademy.com\/wp-content\/uploads\/2026\/08\/ChatGPT-Image-Aug-17-2026-03_35_58-PM-1-1536x728.png 1536w, https:\/\/blog-admin.thethinkacademy.com\/wp-content\/uploads\/2026\/08\/ChatGPT-Image-Aug-17-2026-03_35_58-PM-1-1300x616.png 1300w, https:\/\/blog-admin.thethinkacademy.com\/wp-content\/uploads\/2026\/08\/ChatGPT-Image-Aug-17-2026-03_35_58-PM-1.png 1822w\" sizes=\"auto, (max-width: 1024px) 100vw, 1024px\" \/><\/a><\/figure>\n","protected":false},"excerpt":{"rendered":"<p>The rules in divisibility are a set of shortcuts that tell you whether one number divides &hellip; <a title=\"Rules of Divisibility Explained: Complete Guide with Examples\" class=\"hm-read-more\" href=\"https:\/\/blog-admin.thethinkacademy.com\/blog\/2026\/08\/17\/rules-in-divisibility-guide\/\"><span class=\"screen-reader-text\">Rules of Divisibility Explained: Complete Guide with Examples<\/span>Read more<\/a><\/p>\n","protected":false},"author":1,"featured_media":65456,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[1752,5907,17135,17160,17187],"tags":[],"class_list":["post-65455","post","type-post","status-publish","format-standard","has-post-thumbnail","hentry","category-competitions","category-education","category-study-tips-and-tools","category-math-skills","category-curriculum-by-grade"],"yoast_head":"<!-- This site is optimized with the Yoast SEO plugin v28.4 - https:\/\/yoast.com\/product\/yoast-seo-wordpress\/ -->\n<title>Rules in Divisibility: Guide with Examples and Practice Problems<\/title>\n<meta name=\"description\" content=\"Rules in divisibility \u2014 every rule from 2 to 13, how to prove them, worked examples, practice problems and where they appear in competitions.\" \/>\n<meta name=\"robots\" content=\"index, follow, max-snippet:-1, max-image-preview:large, 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