Problems in Number Theory

Problem:

Solution:

The key step in this problem is to let $n = A2^p3^q5^r$, it can be easily seen that $A = 1$ for minimal value in the set.

The fact that these roots are natural number give rise to these equations:

$p - 1 = 0 \mod 2$
$p = 0 \mod 3$
$p = 0 \mod 5$

$q = 0 \mod 2$
$q - 1 = 0 \mod 3$
$q = 0 \mod 5$

$r = 0 \mod 2$
$r = 0 \mod 3$
$r - 1 = 0 \mod 5$

By Chinese remainder theorem, we get $p = 15$, $q = 10$, $r = 6$, so the answer is

$2^15 \times 3^10 \times 5^6 = 30,233,088,000,000$