Lesson Divisibility by 5 rule

Algebra ->  Divisibility and Prime Numbers -> Lesson Divisibility by 5 rule      Log On


   

This Lesson (Divisibility by 5 rule) was created by by ikleyn(53751) About Me : View Source, Show
About ikleyn:

Divisibility by 5 rule


The  "Divisibility by 5" rule  is as follows:


    An integer number is divisible by  5  if and only if its last digit is  5  or  0.    


Example 1

Check if the number  705  is divisible by  5.

Solution

The last digit of this number is  5.
Hence,  the original number  705  is divisible by  5,  in accordance with the  "Divisibility by 5" rule.

You may check it by making the direct division:  705%2F5 = 141.  It shows that the number  705  is divisible by  5.
The  Divisibility rule  allows you to get the conclusion without making long calculations.


Example 2

Check if the number  371  is divisible by  5.

Solution

The last digit of this number is  1,  which is nor  5  neither  0.
Hence,  the original number  371  is not divisible by  5,  in accordance with the  "Divisibility by 5" rule.


Example 3

Check if the number  790  is divisible by  5.

Solution

The last digit of this number is  0.
Hence,  the original number  790  is divisible by  5,  in accordance with the  "Divisibility by 5" rule.

You may check it by making the direct division:  790%2F5 = 158.  It shows that the number  790  is divisible by  5.
Again,  the  Divisibility rule  allows you to get the conclusion without making long calculations.


The proof of the "divisibility by 5" rule


The proof is very similar to that on the  "divisibility by 2"  rule  (see the lesson  Divisibility by 2 rule  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%5Bn%5Da%5Bn-1%5D . . . a%5B2%5Da%5B1%5Da%5B0%5D  be the decimal record of our integer number  N.
Then  N = a%5Bn%5D%2A10%5En + a%5Bn-1%5D%2A10%5E%28n-1%29 + . . . + 10%5E2%2Aa%5B2%5D + 10%2Aa%5B1%5D + a%5B0%5D.

The first  n  additives,  a%5Bn%5D%2A10%5En,  a%5Bn-1%5D%2A10%5E%28n-1%29, . . . ,  10%5E2%2Aa%5B2%5D  and  10%2Aa%5B1%5D  have the common multiple  10,  which is divisible by  5.
Hence,  the sum of these additives  a%5Bn%5D%2A10%5En + a%5Bn-1%5D%2A10%5E%28n-1%29 + . . . + 10%5E2%2Aa%5B2%5D + 10%2Aa%5B1%5D  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 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

    - Divisibility by 2 rule
    - Divisibility by 3 rule
    - Divisibility by 4 rule
    - Divisibility by 6 rule
    - Divisibility by 9 rule
    - Divisibility by 10 rule
    - Divisibility by 11 rule
    - Restore the omitted digit in a number in a way that the number is divisible by 9
    - Restore the omitted digit in a number in a way that the number is divisible by 11
    - Can there be a perfect square ?
    - Math circle level problems on divisibility numbers
    - Math circle level problem on restoring digit in the product of two 16-digit numbers
    - Math circle level problem on finding remainders
    - OVERVIEW of Divisibility rules by 2, 3, 4, 5, 6, 9, 10 and 11

Lessons that are closely adjacent to these are

    - Product of two consecutive integers is divisible by 2
    - Product of three consecutive integers is divisible by 6
    - Problems dealing with the product of two consecutive integers
    - Problems dealing with the product of three consecutive integers

This lesson has been accessed 5419 times.