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

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

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Given a number `2321`. Is it divisible by `58`?

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

- Yes
- No
- No
- Can't say

The answer is "No". `2321-58xx4 = 1`. So, the number is not divisible by `58`

Consider the divisibility of dividend `2321` by `58`.

An observation is that if a number is multiple of divisor `58`, then the number can be written as `58 xx n`. The same can be written as `2xx29xxn`, because `58=2xx29`.

What does this imply?

- the dividend is a multiple of `2` and `29`
- the dividend is a multiple of `2` and `29`
- this does not imply anything

The answer is "the dividend is a multiple of `2` and `29`". Since `2321` is an odd number, it is not divisible by `2` and so it is not divisible by `58`.

Consider the divisibility of `922` by `58`.

It is observed that both are even numbers. So `922 = 2 xx 461` and `58=2xx29`.

What does this imply?

- the factor `461` should be a multiple of `29` for the divisibility test to pass
- the factor `461` should be a multiple of `29` for the divisibility test to pass
- this does not mean anything

The answer is "the factor `461` should be a multiple of `29` for the divisibility test to pass". That is divisibility test of `922` by `58` is now changed to divisibility test of `461` by `29`. The common factor `2` is removed to simplify.

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

• If the dividend is not divisible by a factor of divisor, then the dividend is not divisible by the composite divisor.

eg: divisibility of `2321` by `58`. `58 = 2xx 29`. The dividend `2321` is not divisible by a factor of divisor `2`, so `2321` is not divisible by `58`.

• If the dividend is divisible by a factor, then both the dividend and divisor can be divided by the factor and the test is done on the results.

eg: Divisibility of `922` by `58`. Both has common factor `2` and so it is simplified as divisibility of `461` by `29`. This is named as **Simplification by Factors**.

**Simplification by Factors**: To simplify divisibility test of dividend by a divisor,

• any common factors can be removed from both the numbers and divisibility test can be performed on simplified dividend by the simplified divisor.

• the dividend has to have all the factors of the divisor to pass the divisibility test.

Check divisibility of `2016` by `264` The figure shows a simplified procedure by which simplification by factors can be performed.

- Divisible
- Not Divisible
- Not Divisible

The answer is "Not divisible"

Check divisibility of `3120` by `156` The figure shows the simplified procedure by which simplification by factors is performed.

- Divisible
- Not Divisible
- Not Divisible

The answer is "Divisible".

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