Find the smallest positive integer n for which 126n cube root is a whole number

Answer 1

any perfect cube has 3 equal factors
e.g. 216 = 6 x 6 x 6

126n = 2 x 3 x 3 x 7 n
so to have triples, we need 2x2, 3, and 7x7
n = 4(3)(49) = 588

our number 126n would be 74088

check:
74088 = (2x2x2)(3x3x3)(7x7x7)