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

Question

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

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Answer 1 · 2017-08-31T22:35:36

1. $61$

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 $61$