Lesson Divisibility by 10 rule
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<H2>Divisibility by 10 rule</H2> The <B><U>"Divisibility by 10" rule</U></B> is as follows: <BLOCKQUOTE><TABLE BORDER=2> <TR> <TD> An integer number is divisible by <B>10</B> if and only if its last digit is <B>0</B>. </TD> </TR> </TABLE></BLOCKQUOTE> <H3>Example 1</H3>Check if the number 705 is divisible by <B>10</B>. <B>Solution</B> The last digit of this number is 5, which not the zero. Hence, the original number 705 is not divisible by <B>10</B>, in accordance with the <B>"Divisibility by 10" rule</B>. You may check that the remainder of the division of 705 by 10 is 5. The <B>Divisibility rule</B> allows you to get the conclusion without making long calculations. <H3>Example 2</H3>Check if the number 371 is divisible by <B>10</B>. <B>Solution</B> The last digit of this number is 1, which is not the zero. Hence, the original number 371 is not divisible by <B>10</B>, in accordance with the <B>"Divisibility by 10" rule</B>. <H3>Example 3</H3>Check if the number 790 is divisible by <B>10</B>. <B>Solution</B> The last digit of this number is 0. Hence, the original number 790 is divisible by <B>10</B>, in accordance with the <B>"Divisibility by 5" rule</B>. You may check it by making the direct division: {{{790/10}}} = {{{79}}}. It shows that the number 790 is divisible by <B>10</B>. The <B>Divisibility rule</B> allows you to get the conclusion without making long calculations. <H3>The proof of the "divisibility by 10" rule</H3> The proof is very similar to that on the "divisibility by 5" rule (see the lesson <A HREF=http://www.algebra.com/algebra/homework/divisibility/lessons/Divisibility-by-5-rule.lesson>Divisibility by 5 rule</A> under the current topic in this site). First, let us consider how the "divisibility by 10" rule can be proved for the concrete number 730, for example. We can write 730 = 7*100 + 3*10 + 0. The first two additives, 7*100 and 3*10, have the common multiple 10. Hence, the sum of these additives 7*100 + 3*10 is divisible by 10. Therefore, the divisibility by 10 of our original number depends on and is determined solely by the last additive, which is the last digit of the number, i.e. 0 in our case. For the general case, the proof follows the same arguments. Let {{{a[n]}}}{{{a[n-1]}}} . . . {{{a[2]}}}{{{a[1]}}}{{{a[0]}}} be the decimal record of our integer number <B>N</B>. Then <B>N</B> = {{{a[n]*10^n}}} + {{{a[n-1]*10^(n-1)}}} + . . . + {{{10^2*a[2]}}} + {{{10*a[1]}}} + {{{a[0]}}}. The first <B>n</B> additives, {{{a[n]*10^n}}}, {{{a[n-1]*10^(n-1)}}}, . . . , {{{10^2*a[2]}}} and {{{10*a[1]}}} have the common multiple 10. Hence, the sum of these additives {{{a[n]*10^n}}} + {{{a[n-1]*10^(n-1)}}} + . . . + {{{10^2*a[2]}}} + {{{10*a[1]}}} is divisible by <B>10</B>. Therefore, the divisibility by <B>10</B> of our original number depends on and is determined solely by the last additive, which is presented by the last digit of the number. It is what has to be proved. My other lessons in this site on divisibility rules are - <A HREF=http://www.algebra.com/algebra/homework/divisibility/lessons/Divisibility-by-2-rule.lesson>Divisibility by 2 rule</A> - <A HREF=http://www.algebra.com/algebra/homework/divisibility/lessons/Divisibility-by-3-rule.lesson>Divisibility by 3 rule</A> - <A HREF=http://www.algebra.com/algebra/homework/divisibility/lessons/Divisibility-by-4-rule.lesson>Divisibility by 4 rule</A> - <A HREF=http://www.algebra.com/algebra/homework/divisibility/lessons/Divisibility-by-5-rule.lesson>Divisibility by 5 rule</A> - <A HREF=http://www.algebra.com/algebra/homework/divisibility/lessons/Divisibility-by-6-rule.lesson>Divisibility by 6 rule</A> - <A HREF=http://www.algebra.com/algebra/homework/divisibility/lessons/Divisibility-by-9-rule.lesson>Divisibility by 9 rule</A> - <A HREF=http://www.algebra.com/algebra/homework/divisibility/lessons/Divisibility-by-11-rule.lesson>Divisibility by 11 rule</A> - <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> - <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> - <A HREF=https://www.algebra.com/algebra/homework/divisibility/lessons/Can-there-be-a-perfect-square.lesson>Can there be a perfect square ?</A> - <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> - <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> - <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> - <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 - <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> - <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> - <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> - <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>
