Show that if a and b are positive integers and a^3 | b^3, then a | b

If you're allowed to use prime factorisation then it's fairly easy:

Let:
b = p1^b1 * p2^b2 * ... * pn^bn
a = p1^a1 * p2^a2 * ... * pn^an
where p1.. pn are primes, a1... an and b1...bn are non negative integers
=>
b^3 = p1^(3b1) * p2^(3b2) * ... * pn^(3bn)
a^3 = p1^(3a1) * p2^(3a2) * ... * pn^(3an)

a^3 | b^3, so 3ai <= 3bi for all i
which means ai <= bi for all i
which means a | b