__maths__>__Divisibility in Whole Numbers__>__Simplification of Divisibility Tests: 8, 12, 15__### Divisibility: Simplification by Division

In this page, Simplification of divisibility test by division is explained. This is used to develop other divisibility tests for specific numbers.

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Given a number `340`. is it divisible by `21`?

Check this by long division method or by simplification by subtraction.

- Yes
- No
- No
- Can't say

The answer is "No". `340-210=130` and `130` is not divisible by `21`. So, the number is not divisible by `21`

Consider divisibility of `340` by `21`.

An observation is that the dividend is an even number and the divisor is not. The dividend can be written as `2xx170`.

Which of the following is true?

- the divisibility test can be performed on `170` as the factor `2` is a co-prime of `21`
- the divisibility test can be performed on `170` as the factor `2` is a co-prime of `21`
- the divisibility test cannot be simplified

The answer is "the divisibility test can be performed on `170` as the factor `2` is co-prime of `21`".

This can further be simplified as `170-:2-:5 = 17`, as `2` and `5` are co-primes of `21`.

To check the divisibility of a composite dividend by a divisor, the factors of the dividend can help in simplifying the divisibility test.

• If a factor of dividend is co-prime of divisor, then the dividend can be divided by the co-prime factor and the divisibility test can be performed on the result.

This is named as **Simplification by Division**.

**Simplification by Division**: Dividend can be divided by a factor, that is co-prime of divisor, and the result is checked for divisibility.

Check divisibility of `990` by `13`

- Divisible
- Not Divisible
- Not Divisible

The answer is "Not divisible". `10` and `9` are co-primes of `13`. So the number is divided as `990-:90 = 11`. The result `11` is not divisible by `13`.

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