Each of X and Y is a positive integer which is not a perfect square that satisfy:

√X - √Y = √17

Is each of X and Y always divisible by 17?

Give reasons for your answer.

sqrt(x) = sqrt(17) + sqrt(y)

squaring gives

x = 17 + y + 2sqrt(17y)

the LHS is integer. so the RHS is also, so y must be a multiple of 17

Similarly.

sqrt(y) = -sqrt(17) + sqrt(x)

squaring gives

y = 17 + x - 2sqrt(17x)

the LHS is integer. so the RHS is also, so x must be a multiple of 17

Note that this is true for any x and y, even if x or y is a perfect square