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263 lines
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HTML
263 lines
9.8 KiB
HTML
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<article id="content" role="main">
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<h1 class="center">Divisibility Rules</h1>
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<p class="center"><b>Easily test if one number can be exactly divided by another</b></p>
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<h2>Divisible By</h2>
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<p>"Divisible By" means "when you divide one number by another the result is a <a href="whole-numbers.html">whole number</a>"</p>
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<div class="example">
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<h3>Examples:</h3>
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<p>14 <b>is</b> divisible by 7, because 14 ÷ 7 = 2 <b>exactly</b></p>
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<p>15 is <b>not</b> divisible by 7, because 15 ÷ 7 =<b> 2 <span class="intbl"><em>1</em><strong>7</strong></span></b> (the result is <b>not</b> a whole number)</p>
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<p>0 <b>is</b> divisible by 7, because 0 ÷ 7 = 0 <b>exactly</b> (0 is a whole number)</p>
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</div>
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<div class="words">"Divisible by" and "can be exactly divided by" mean the same thing</div>
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<h2>The Divisibility Rules</h2>
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<p>These rules let you test if one number is divisible by another, without having to do too much calculation!</p>
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<div class="example">
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<h3>Example: is 723 divisible by 3?</h3>
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<p>We could try dividing 723 by 3</p>
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<p>Or use the "3" rule: 7+2+3=12, and 12 ÷ 3 = 4 exactly <b>Yes</b></p>
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</div>
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<p><i>Note: Zero is divisible by <b>any number</b> (except by itself), so gets a "yes" to all these tests.</i></p>
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<p> </p>
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<div class="def">
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<div class="wow1">1</div>
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<p>Any integer (not a fraction) is divisible by 1</p><br>
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</div>
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<div class="def">
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<div class="wow1">2</div>
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<p>The last digit is even (0,2,4,6,8)</p>
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<p class="so">12<b>8</b> <b>Yes</b></p>
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<p class="so">12<b>9</b> <b>No</b></p>
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</div>
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<div class="def">
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<div class="wow1">3</div>
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<p>The sum of the digits is divisible by 3
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</p>
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<p class="so">381 (3+8+1=12, and 12÷3 = 4) <b>Yes</b></p>
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<p class="so">217 (2+1+7=10, and 10÷3 = 3 <span class="frac"><sup>1</sup>/<sub>3</sub></span>) <b>No</b></p>
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<p>This rule can be repeated when needed:</p>
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<p class="so">99996 (9+9+9+9+6 = 42, then 4+2=6) <b>Yes</b>
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</p></div><p></p>
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<div class="def">
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<div class="wow1">4</div>
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<p>The last 2 digits are divisible by 4</p>
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<p class="so">13<b>12</b> is (12÷4=3) <b>Yes</b></p>
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<p class="so">70<b>19</b> is not (19÷4=4 <span class="frac"><sup>3</sup>/<sub>4</sub></span>) <b>No</b></p>
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<p>A quick check (useful for small numbers) is to halve the number twice and the result is still a whole number.</p>
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<p class="so">12/2 = 6, 6/2 = 3, 3 is a whole number. <b>Yes</b></p>
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<p class="so">30/2 = 15, 15/2 = 7.5 which is not a whole number. <b>No</b></p>
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</div><p></p>
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<div class="def">
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<div class="wow1">5</div>
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<p>The last digit is 0 or 5</p>
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<p class="so">17<b>5</b> <b>Yes</b></p>
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<p class="so">80<b>9</b> <b>No</b></p>
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</div>
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<div class="def">
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<div class="wow1">6</div>
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<p>Is even and is divisible by 3 (it passes both the 2 rule and 3 rule above)</p>
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<p class="so">114 (it is even, and 1+1+4=6 and 6÷3 = 2) <b>Yes</b></p>
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<p class="so">308 (it is even, but 3+0+8=11 and 11÷3 = 3 <span class="frac"><sup>2</sup>/<sub>3</sub></span>) <b>No</b></p>
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</div>
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<div class="def">
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<div class="wow1">7</div>
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<p>Double the last digit and subtract it from a number made by the other digits. The result must be divisible by 7.
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(We can apply this rule to that answer again)
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</p>
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<p class="so">672 (Double 2 is 4, 67−4=63, and 63÷7=9) <b>Yes</b></p>
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<p class="so">105 (Double 5 is 10, 10−10=0, and 0 is divisible by 7) <b>Yes</b></p>
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<p class="so">905 (Double 5 is 10, 90−10=80, and 80÷7=11 <span class="frac"><sup>3</sup>/<sub>7</sub></span>) <b>No</b></p>
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</div><p></p>
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<div class="def">
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<div class="wow1">8</div>
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<p>The last three digits are divisible by 8
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</p>
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<p class="so">109<b>816</b> (816÷8=102) <b>Yes</b></p>
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<p class="so">216<b>302</b> (302÷8=37 <span class="frac"><sup>3</sup>/<sub>4</sub></span>) <b>No</b></p>
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<p>A quick check is to halve three times and the result is still a whole number:</p>
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<p class="so">816/2 = 408, 408/2 = 204, 204/2 = 102 <b>Yes</b></p>
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<p class="so">302/2 = 151, 151/2 = 75.5 <b>No</b></p>
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</div><p></p>
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<div class="def">
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<div class="wow1">9</div>
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<p>The sum of the digits is divisible by 9<br>
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<br>
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(Note: This rule can be repeated when needed)
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</p><p class="so">1629 (1+6+2+9=18, and again, 1+8=9) <b>Yes</b></p>
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<p class="so">2013 (2+0+1+3=6) <b>No</b></p>
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</div><p></p>
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<div class="def">
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<div class="wow1">10</div>
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<p>The number ends in 0</p>
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<p class="so">22<b>0</b> <b>Yes</b></p>
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<p class="so">22<b>1</b> <b>No</b></p>
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</div><p></p>
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<div class="def">
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<div class="wow1">11</div>
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<p>Add and subtract digits in an alternating pattern (add digit, subtract next digit, add next digit, etc). Then check if that answer is divisible by 11.</p>
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<p class="so"><b>1</b>3<b>6</b>4 (+1−3+6−4 = <b>0</b>) <b>Yes</b></p>
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<p class="so"><b>9</b>1<b>3</b> (+9−1+3 = <b>11</b>) <b>Yes</b></p>
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<p class="so"><b>3</b>7<b>2</b>9 (+3−7+2−9 = <b>−11</b>) <b>Yes</b></p>
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<p class="so"><b>9</b>8<b>7</b> (+9−8+7 = <b>8</b>) <b>No</b></p>
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</div><p></p>
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<div class="def">
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<div class="wow1">12</div>
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<p>The number is divisible by both 3 <i><b>and</b></i> 4
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(it passes both the 3 rule and 4 rule above)
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</p>
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<p class="so">648<br>
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(<i>By 3?</i> 6+4+8=18 and 18÷3=6 Yes)<br>
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<i>(By 4?</i> 48÷4=12 Yes)<br>
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Both pass, so <b>Yes</b></p>
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<p class="so">524<br>
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(<i>By 3?</i> 5+2+4=11, 11÷3= 3 <span class="frac"><sup>2</sup>/<sub>3</sub></span> No)<br>
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(Don't need to check by 4) <b>No</b></p>
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</div><p></p>
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<div class="simple"> </div>
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<p>There are lots more! Not only are there divisibility tests for larger numbers, but there are more tests for the numbers we have shown.</p>
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<h2>Factors Can Be Useful</h2>
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<p>Factors are the numbers you multiply to get another number:</p>
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<p class="center"><img src="numbers/images/factor-2x3.svg" alt="factors" height="70" width="181"></p>
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<p>This can be useful, because:</p>
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<p class="center"><span class="larger">When a number is divisible by another number ...</span></p>
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<p class="larger" align="right">... then it is <b>also</b> divisible by each of the factors of that number.</p>
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<p> </p>
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<div class="example">
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<p>Example: If a number is divisible by 6, it is also divisible by 2 and 3</p>
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</div>
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<div class="example">
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<p>Example: If a number is divisible by 12, it is also divisible by 2, 3, 4 and 6</p>
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</div>
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<h2>Another Rule For 11</h2>
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<ul>
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<li>Subtract the last digit from a number made by the other digits.</li>
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<li>If that number is divisible by 11 then the original number is, too.</li>
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</ul>
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<p>Can repeat this if needed,</p>
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<div class="example">
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<h3>Example: 286</h3>
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<p>28 − 6 is 22, which <b>is</b> divisible by 11, so 286 is divisible by 11</p>
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</div>
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<div class="example">
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<h3>Example: 14641</h3>
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<ul>
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<li>1464 − 1 is 1463</li>
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<li>146 − 3 is 143</li>
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<li>14 − 3 is 11, which <b>is</b> divisible by 11, so 14641 is divisible by 11</li>
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</ul>
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</div>
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<p> </p>
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<div class="questions">1625, 1626, 1627, 1628, 2689, 3599, 3600, 3601, 3602, 5007</div>
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<div class="related">
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<a href="fractions_division.html">Division of Fractions</a>
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<a href="dividing-decimals.html">Dividing Decimals</a>
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<a href="numbers/index.html">Numbers Index</a>
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</div>
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