If y is an integer, how many distinct integer values are possible for

Rephrase the question: this is like asking to fill in the blank in "\(2^{30}\cdot3^{24}\cdot5^{48}\) is a perfect ____ power"

A number is a perfect \(k^{th}\) power if the exponents in the prime factorization of the number are all multiples of \(k\).

So, this question is the same as asking to fill in the blank in "\(30, 24,\) and \(48\) are all divisible by ____"

To answer this question, we can find the GCF of \(30, 24,\) and \(48\) ... which is \(6\).

If the GCF of a set of numbers is \(6\), then each number will be divisible by any factor of \(6\).

So, \(n\) could be \(1, 2, 3,\) and \(6\). Answer D.

For example:

\((2^{30}\cdot3^{24}\cdot5^{48})^1=2^{30}\cdot3^{24}\cdot5^{48}\)

\((2^{15}\cdot3^{12}\cdot5^{24})^2=2^{30}\cdot3^{24}\cdot5^{48}\)

\((2^{10}\cdot3^{8}\cdot5^{16})^3=2^{30}\cdot3^{24}\cdot5^{48}\)

\((2^{5}\cdot3^{4}\cdot5^{8})^6=2^{30}\cdot3^{24}\cdot5^{48}\)