Find all three-digit even numbers N such that 693xN is a perfect square, that is, 693 x N = k2 where k is an interger.

Answer 1

Use the fact that any two perfect squares multiplied will produce another perfect square.
e.g. 4x9 = 36

693 = 9x7x11 so we would need multipliers of 7 and 11 to yield a perfect square

so 693x7x11 = 9x49x121

so N could be 77, but you wanted N to be even.
any number multiplied by an even will yield an even.

so N = 77x(any even perfect square)

e.g.
N = 77x4 or
N = 77x16 etc

Answer 2

308, yes exactly, I see.
Thank-you very much.

Answer 3

can you plz explain
693 = 9x7x11 so we would need multipliers of 7 and 11 to yield a perfect square

so 693x7x11 = 9x49x121
Thanks