__maths__>__Divisibility in Whole Numbers__>__Reducing number of Digits for Divisibility Test__### Divisibility: Simplification in Digits

In this page, Simplification of Divisibility test, by reducing the number of digits of the dividend, is explained. This method is used to derive divisibility tests for `7` and `13`.

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What is "Divisibility Test: Simplification by Subtraction"?

- difference between the dividend and a multiple of divisor is checked for divisibility by divisor
- difference between the dividend and a multiple of divisor is checked for divisibility by divisor
- dividend and divisor are divided by common factors and the result is checked for divisibility

The answer is "difference between the dividend and a multiple of divisor is checked for divisibility by divisor"

What is "Divisibility Test: Simplification by Division"?

- dividend can be divided by a co-prime (of divisor) factor and result is checked for divisibility
- dividend and divisor are divided by common factors and the result is checked for divisibility
- both the above
- both the above

The answer is "both the above".

The "simplification by subtraction" and "simplification by division" are further refined.

Take a number `2352` and check for divisibility by `7`.

We can simplify the number as

divisibility of `2352`*subtracting a multiple of `7` *

divisibility `->2352-7xx3xx2`

divisibility `->2352-42 = 2310` *`2` and `5` are co-primes of divisor `7`*

divisibility `->2310/10 = 231`* repeating subtraction of a multiple of `7`*

divisibility `->231-7xx3xx1 = 210` * repeating `2` and `5` co-prime simplification*

divisibility `->210/10 = 21`

`21` is divisible by `7`, and so `2352` is divisible by `7`.

Note: The multiple of `7` is chosen with two factors

• First `3` is chosen because, `7xx3 = 21` where the units place is `1`

• Second `2` is chosen because the units digit of the number is `2` and on subtraction that makes the units digit `0`. That is, `2352-21*2 = 2310`. This facilitates the co-prime of `10` step, thereby reducing the number of digits.

Another example of combining "simplification by subtraction" and "simplification by division".

Take a number `2352` and check for divisibility by `13`.

Look at the multiples of `13` : `26`, `39`, etc.

The multiple `39` is interesting as `39*n+n` is a multiple of `10`.

We can simplify the number `2352` as

divisibility of `2352`*adding a multiple of `13` *

divisibility `->2352+13xx3xx2`

divisibility `->2352+78 = 2430` *`2` and `5` are co-primes of divisor `13`*

divisibility `->2430/10 = 243`* repeating addition of a multiple of `13`*

divisibility `->231+13xx3xx1 = 270` * repeating `2` and `5` co-prime simplification*

divisibility `->270/10 = 27`

`27` is not divisible by `13`, and so `2352` is not divisible by `13`.

Note: The multiple of `13` is chosen with two factors

• First `3` is chosen because, `13xx3 = 39` where the units place is `9`

• Second `2` is chosen because the units digit of the number is `2` and on addition that makes the units digit `0`. That is, `2352+39*2 = 2430`. This facilitates the co-prime of `10` step, thereby reducing the number of digits.

**Simplification in Digits** : Consider the multiples of the divisor and choose the one that ends `1` or `9`. By using Simplification by Subtraction and Simplification by Division, reduce the number of digits of the dividend. The reduced dividend can be checked for divisibility, and the simplification can be used repeatedly.

*slide-show version coming soon*