The other tutors did not mention that there are two answers.
The problem did not state that k must be a POSITIVE integer.  It
just stated that k must be an integer, and an integer can be
POSITIVE or NEGATIVE.  So we must give both answers.

If 64x² + kxy + y²

is a perfect square it will have to factor as

either  (8x + y)² or as (8x - y)²

If it factors as (8x + y)², then that is the same as

(8x + y)(8x + y) = 64x² + 16xy + y².  In that case the

middle term +kxy would equal to +16xy and so k would be 16.

-----------------------------------------

If it factors as (8x - y)², then that is the same as

(8x - y)(8x - y) = 64x² - 16xy + y².  In that case the

middle term +kxy would equal to -16xy and so k would be -16.

Answer: Two possibilities: k = 16, k = -16

Edwin