__maths__>__Divisibility in Whole Numbers__>__Reducing number of Digits for Divisibility Test__### Divisibility by `13`

In this page, a simple overview of the divisibility test for `13` is provided. The procedure uses simplification in Digits.

*click on the content to continue..*

Which of the following helps in divisibility test of `13`?

- simplification in digits as `13xx3=39`
- simplification in digits as `13xx3=39`
- divisibility test cannot be simplified for `13`

The answer is "simplification in digits as `13xx3=39`".

If the given dividend is `10A+B`, (where `B` is a single digit number)

then the dividend can be modified to `10A+B+39B` *(simplification by addition)*

which equals `10A+40B = 10(A+4B)`.

Since `13` is co-prime to both `2` and `5`, the divisibility test is done on `A+4B` *(simplification by division)*. This, in effect, reduced the number of digits in `10A+B`.

Note: `A+4B` can further be simplified using the same procedure.

**Test for Divisibility by `13`** : Reduce the number of digits by iteration.

Remove the units digit from the dividend to get a modified number. Add `4` times of the removed units digit to the modified number.

Perform the divisibility test on the sum.

Is `2008` divisible by `13`?

- Yes
- No
- No

The answer is "No". Following the process

`2008`

`->200+4xx8`

`->232`

`->23+4xx2`

`->31`

`31` is not divisible by `13`, and so it is concluded that the number is not divisible by `13`.

Is `416` divisible by `13`?

- Yes
- Yes
- No

The answer is "Yes". Following the process

`416`

`->41+4xx6`

`->65`

`->6+4xx5`

`->26`

`26` is divisible by `13`, and so it is concluded that the number is divisible by `13`.

*slide-show version coming soon*