Question

The number `107^90 - 76^90` is divisible by ?

options :-
1. 61
2. 62
3. 64
4. none of these

Answer 1

1. `61`

Explanation:

Given:

`107^90-76^90`

First note that `107^90` is odd and `76^90` is even.

So their difference is odd and cannot be divisible by `62` or `64`.

To check for divisibility by `61`, let us look at powers of `107` and `76` modulo `61`.

`107^1 -= 46`

`107^2 -= 46^2 -= 2116 -= 42`

`76^1 -= 15`

`76^2 -= 15^2 -= 225 -= 42`

So:

`107^2-76^2 -= 0` modulo `61`

That is `107^2-76^2` is divisible by `61`

Then:

`107^90-76^90`

`= (107^2-76^2)(107^88+107^86*76^2+107^84*76^4+...+76^88)`

So:

`107^90-76^90`

is divisible by `61`