2022/063) How can I find all positive integer values of N such that N(N−101) is a square of a positive integer

 Now let us take GCD(N,N-101)

GCD(N,N=101) = GCD(N,101)

There are 2 cases either N is multiple of 101 or not

If N is multiple of 101 say N= 101m

So N-101 = 101(m-1) so $N(N-101) = 101^2m(m-1)$ which is not a perfect square

So N is not a multiple of 101 so GCD(N,N-101) = 1 so N and N-101 both are perfect squares

Say $N= x^2$ and $N-101$ = y^2

$x^2-y^2 = 101$ or $(x+y)(x-y) = 101$ 

So x + y = 101 and x-y =1 solving these we get x = 51 and y = 50

So $N = x^2 = 51^ = 2601$