Question

write the set of all integers whose cube it an even integer​

Answer 1

Any number n is expressible as a product of powers of prime numbers, its prime factors:

n=p1^i1*p2^i2*……….*pk^ik

Thus n is even iff one of its prime factors is 2 say p1=2. Thus its cube is

n^3=p1^(3*i1)*p2^(3*i2*)*……….*pk^(3*ik)

This in turn is even iff one of its prime factors, the same as n itself, is 2 say p1=2. We conclude that n is even iff n^3 is even and thus the set of all integers whose cube is even is the set of even integers!