Question

Some positive integer, x, results in a square number when 64 is taken away from it, and when 25 is added to it. Find x.

Answer 1

x=2000

Explanation:

The conditions state, being x, N, M positive integers:

`x-64=N^2` [1]

`x+25=M^2` [2]

Subtracting the [1] from [2]

`89=M^2-N^2=(M-N)(M+N)`

Since 89 has no other prime factors except 1 and itself, we can write

`(M-N)(M+N)=(1)(89)`

`M+N=89`

`M-N=1`

This system of equations results in
`M=45, N=44`

Replacing in [1] or [2] results

`x=2000`

Proof:

`2000-64=1936 =44^2`

`2000+25=2025=45^2`