What is the smallest integer k such that 756k is a perfect cube?

k = 98.

In the prime factorization (in power format) of a perfect cube, every prime must be to the power of a multiple of 3.

756 = 2^2 x 3^3 x 7

Thus the smallest perfect cube that is a multiple of 756 is 2^3 × 3^3 × 7^3; to obtain this need to multiply 756 by 2^1 × 3^0 × 7^2 = 98

Thus the smallest k to make 756k a perfect prime is k = 98.