Lesson Divisibility by 5 rule
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<H2>Divisibility by 5 rule</H2> The <B><U>"Divisibility by 5" rule</U></B> is as follows: <BLOCKQUOTE><TABLE BORDER=2> <TR> <TD> An integer number is divisible by <B>5</B> if and only if its last digit is <B>5</B> or <B>0</B>. </TD> </TR> </TABLE></BLOCKQUOTE> <H3>Example 1</H3>Check if the number 705 is divisible by <B>5</B>. <B>Solution</B> The last digit of this number is 5. Hence, the original number 705 is divisible by <B>5</B>, in accordance with the <B>"Divisibility by 5" rule</B>. You may check it by making the direct division: {{{705/5}}} = {{{141}}}. It shows that the number 705 is divisible by <B>5</B>. 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>5</B>. <B>Solution</B> The last digit of this number is 1, which is nor <B>5</B> neither 0. Hence, the original number 371 is not divisible by <B>5</B>, in accordance with the <B>"Divisibility by 5" rule</B>. <H3>Example 3</H3>Check if the number 790 is divisible by <B>5</B>. <B>Solution</B> The last digit of this number is 0. Hence, the original number 790 is divisible by <B>5</B>, in accordance with the <B>"Divisibility by 5" rule</B>. You may check it by making the direct division: {{{790/5}}} = {{{158}}}. It shows that the number 790 is divisible by <B>5</B>. Again, the <B>Divisibility rule</B> allows you to get the conclusion without making long calculations. <H3>The proof of the "divisibility by 5" rule</H3> The proof is very similar to that on the "divisibility by 2" rule (see the lesson <A HREF=http://www.algebra.com/algebra/homework/divisibility/lessons/Divisibility-by-2-rule.lesson>Divisibility by 2 rule</A> under the current topic in this site). First, let us consider how the "divisibility by 5" rule can be proved for the concrete number 735, for example. We can write 735 = 7*100 + 3*10 + 5. The first two additives, 7*100 and 3*10, have the common multiple 10, which is divisible by 5. Hence, the sum of these additives 7*100 + 3*10 is divisible by 5. Therefore, the divisibility by 5 of our original number depends on and is determined solely by the last additive, which is the last digit of the number, i.e. 5 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, which is divisible by <B>5</B>. 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>5</B>. Therefore, the divisibility by <B>5</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-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-10-rule.lesson>Divisibility by 10 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>
