Lesson Divisibility by 10 rule

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Divisibility by 10 rule

The "Divisibility by 10" rule is as follows:

An integer number is divisible by 10 if and only if its last digit is 0.

Example 1

Check if the number 705 is divisible by 10.

Solution

The last digit of this number is 5, which not the zero.
Hence, the original number 705 is not divisible by 10, in accordance with the "Divisibility by 10" rule.

You may check that the remainder of the division of 705 by 10 is 5.
The Divisibility rule allows you to get the conclusion without making long calculations.

Example 2

Check if the number 371 is divisible by 10.

Solution

The last digit of this number is 1, which is not the zero.
Hence, the original number 371 is not divisible by 10, in accordance with the "Divisibility by 10" rule.

Example 3

Check if the number 790 is divisible by 10.

Solution

The last digit of this number is 0.
Hence, the original number 790 is divisible by 10, in accordance with the "Divisibility by 5" rule.

You may check it by making the direct division: 790%2F10

. It shows that the number 790 is divisible by 10.
The Divisibility rule allows you to get the conclusion without making long calculations.

The proof of the "divisibility by 10" rule

The proof is very similar to that on the "divisibility by 5" rule (see the lesson Divisibility by 5 rule 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%5Bn%5D