Lesson Divisibility by 6 rule

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<H2>Divisibility by 6 rule</H2>

An integer number is divisible by &nbsp;6&nbsp; if and only if it is divisible by &nbsp;2&nbsp; and by &nbsp;3. &nbsp;By combining the rules of divisibility by &nbsp;2&nbsp; and by &nbsp;3&nbsp; from the lessons &nbsp;<A HREF=http://www.algebra.com/algebra/homework/divisibility/lessons/Divisibility-by-2-rule.lesson>Divisibility by 2 rule</A>&nbsp; and &nbsp;<A HREF=http://www.algebra.com/algebra/homework/divisibility/lessons/Divisibility-by-3-rule.lesson>Divisibility by 3 rule</A>&nbsp; under the current topic in this site, &nbsp;we come to the &nbsp;"divisibility by 6" rule.

The &nbsp;"Divisibility by 6" rule&nbsp; is as follows:

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 &nbsp;&nbsp;&nbsp;&nbsp;An integer number is divisible by &nbsp;6&nbsp; if and only if its last digit is even and the sum of the digits is divisible by &nbsp;3.&nbsp;&nbsp;&nbsp;&nbsp;

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In other words, &nbsp;for checking if the given integer number is divisible by &nbsp;6, &nbsp;make the following steps:

&nbsp;&nbsp;&nbsp;&nbsp;1. &nbsp;Check if the last digit is even.

&nbsp;&nbsp;&nbsp;&nbsp;2. &nbsp;Calculate the sum of the digits of the given number and check if this sum is divisible by &nbsp;3.

&nbsp;&nbsp;&nbsp;&nbsp;3. &nbsp;If the last digit is even and the sum of the digits is divisible by &nbsp;3, &nbsp;then the original number is divisible by &nbsp;6.

&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;If the last digit is not even OR the sum of the digits is not divisible by &nbsp;3,&nbsp; then the original number is not divisible by &nbsp;6.

Let us consider examples.

<H3>Example 1</H3>Check if the number &nbsp;576&nbsp; is divisible by &nbsp;6.

Solution

The last digit &nbsp;6&nbsp; is even.

The sum of the digits of the given number is &nbsp;5 + 7 + 6 = 18. &nbsp;It is divisible by &nbsp;3.

Hence, &nbsp;the original number &nbsp;576&nbsp; is divisible by &nbsp;6, &nbsp;in accordance with the &nbsp;"Divisibility by 6" rule.

You may check it by making the direct division: &nbsp;{{{576/6}}} = {{{96}}}. &nbsp;It shows that the number &nbsp;376&nbsp; is divisible by &nbsp;6.

The &nbsp;Divisibility rule&nbsp; allows you to get the same conclusion without making long calculations.

<H3>Example 2</H3>Check if the number &nbsp;728&nbsp; is divisible by &nbsp;6.

Solution

The last digit &nbsp;8&nbsp; is even.

The sum of the digits of the given number is &nbsp;7 + 2 + 8 = 17. &nbsp;It is not divisible by &nbsp;3.

Hence, &nbsp;the original number &nbsp;728&nbsp; is not divisible by &nbsp;6, &nbsp;in accordance with the &nbsp;"Divisibility by 3" rule.

Below are even more impressive examples.

<H3>Example 3</H3>Check if the number &nbsp;444,444,444&nbsp; is divisible by &nbsp;6.

Solution

The last digit &nbsp;4&nbsp; is even.

The sum of the digits of the given number is &nbsp;4 + 4 + 4 + 4 + 4 + 4 + 4 + 4 + 4 = 36.

It is divisible by &nbsp;3.

Hence, &nbsp;the original number &nbsp;444,444,444&nbsp; is divisible by &nbsp;6, &nbsp;in accordance with the &nbsp;"Divisibility by 6" rule.

Again, the &nbsp;Divisibility rule&nbsp; allows you to get the conclusion without making long calculations.

<H3>Example 4</H3>Check if the number &nbsp;4,444,444,444&nbsp; is divisible by &nbsp;6.

Solution

The last digit &nbsp;4&nbsp; is even.

The sum of the digits of the given number is &nbsp;4 + 4 + 4 + 4 + 4 + 4 + 4 + 4 + 4 + 4 = 40.

It is not divisible by &nbsp;3.

Hence, &nbsp;the original number &nbsp;4,444,444,444&nbsp; is not divisible by &nbsp;6, &nbsp;in accordance with the &nbsp;"Divisibility by 6" rule.

The &nbsp;"Divisibility by 6" rule&nbsp; is simply the combination of the &nbsp;"divisibility by 2" rule&nbsp; and the &nbsp;"divisibility by 3" rule&nbsp; that there were proved in the lessons mentioned above.

It does not require an additional proof.

My other lessons in this site on divisibility rules are

&nbsp;&nbsp;&nbsp;&nbsp;- <A HREF=http://www.algebra.com/algebra/homework/divisibility/lessons/Divisibility-by-2-rule.lesson>Divisibility by 2 rule</A>

&nbsp;&nbsp;&nbsp;&nbsp;- <A HREF=http://www.algebra.com/algebra/homework/divisibility/lessons/Divisibility-by-3-rule.lesson>Divisibility by 3 rule</A>

&nbsp;&nbsp;&nbsp;&nbsp;- <A HREF=http://www.algebra.com/algebra/homework/divisibility/lessons/Divisibility-by-4-rule.lesson>Divisibility by 4 rule</A>

&nbsp;&nbsp;&nbsp;&nbsp;- <A HREF=http://www.algebra.com/algebra/homework/divisibility/lessons/Divisibility-by-5-rule.lesson>Divisibility by 5 rule</A>

&nbsp;&nbsp;&nbsp;&nbsp;- <A HREF=http://www.algebra.com/algebra/homework/divisibility/lessons/Divisibility-by-9-rule.lesson>Divisibility by 9 rule</A>

&nbsp;&nbsp;&nbsp;&nbsp;- <A HREF=http://www.algebra.com/algebra/homework/divisibility/lessons/Divisibility-by-10-rule.lesson>Divisibility by 10 rule</A>

&nbsp;&nbsp;&nbsp;&nbsp;- <A HREF=http://www.algebra.com/algebra/homework/divisibility/lessons/Divisibility-by-11-rule.lesson>Divisibility by 11 rule</A>

&nbsp;&nbsp;&nbsp;&nbsp;- <A HREF=http://www.algebra.com/algebra/homework/divisibility/lessons/Restore-the-omitted-digit-in-a-number-in-a-way-the-number-is-divisible-by-9.lesson>Restore the omitted digit in a number in a way that the number is divisible by 9</A>

&nbsp;&nbsp;&nbsp;&nbsp;- <A HREF=http://www.algebra.com/algebra/homework/divisibility/lessons/Restore-the-omitted-digit-in-a-number-in-a-way-the-number-is-divisible-by-11.lesson>Restore the omitted digit in a number in a way that the number is divisible by 11</A>

&nbsp;&nbsp;&nbsp;&nbsp;- <A HREF=https://www.algebra.com/algebra/homework/divisibility/lessons/Can-there-be-a-perfect-square.lesson>Can there be a perfect square ?</A>

&nbsp;&nbsp;&nbsp;&nbsp;- <A HREF=https://www.algebra.com/algebra/homework/divisibility/lessons/Math-circle-level-problems-on-divisibility-numbers.lesson>Math circle level problems on divisibility numbers</A>

&nbsp;&nbsp;&nbsp;&nbsp;- <A HREF=https://www.algebra.com/algebra/homework/divisibility/lessons/Math-circle-level-problem-on-restoring-digit-in-the-product-of-two-16-digit-numbers.lesson>Math circle level problem on restoring digit in the product of two 16-digit numbers</A>

&nbsp;&nbsp;&nbsp;&nbsp;- <A HREF=https://www.algebra.com/algebra/homework/divisibility/lessons/Math-circle-level-problem-on-finding-remainders.lesson>Math circle level problem on finding remainders</A>

&nbsp;&nbsp;&nbsp;&nbsp;- <A HREF=http://www.algebra.com/algebra/homework/divisibility/lessons/OVERVIEW-of-Divisibility-rules-by-2-3-4-5-6-9-10-11.lesson>OVERVIEW of Divisibility rules by 2, 3, 4, 5, 6, 9, 10 and 11</A>

Lessons that are closely adjacent to these are

&nbsp;&nbsp;&nbsp;&nbsp;- <A HREF=http://www.algebra.com/algebra/homework/divisibility/lessons/Product-of-two-consecutive-integers-is-divisible-by-2.lesson>Product of two consecutive integers is divisible by 2</A>,

&nbsp;&nbsp;&nbsp;&nbsp;- <A HREF=http://www.algebra.com/algebra/homework/divisibility/lessons/Product-of-three-consecutive-integers-is-divisible-by-6.lesson>Product of three consecutive integers is divisible by 6</A>

&nbsp;&nbsp;&nbsp;&nbsp;- <A HREF=http://www.algebra.com/algebra/homework/Problems-with-consecutive-odd-even-integers/Problems-dealing-with-the-product-of-two-consecutive-integers.lesson>Problems dealing with the product of two consecutive integers</A>

&nbsp;&nbsp;&nbsp;&nbsp;- <A HREF=http://www.algebra.com/algebra/homework/Problems-with-consecutive-odd-even-integers/Problems-dealing-with-the-product-of-three-consecutive-integers.lesson>Problems dealing with the product of three consecutive integers</A>