If x is a positive integer, is x a multiple of 12?

Bunuel: Could you explain the concept behind why X= 2^3*3 for X^2=2^5*3.

bb can you please help me out with this problem?

Stat (1)
This is a tricky problem. You need to think about prime factorization. Find the P.F. of 96.

> \(96 = 2^5 × 3^1\)

When you square something, the exponents on the prime factors are multiplied by 2, so they will be even. Thus, the exponents of \(x^2\) must be even. You need to bump the exponents up to the next even number.

> \(x^2\) is a multiple of \(2^6 × 3^2\)

> \(x\) is a multiple of \(2^3 × 3^1 = 24\)

In other words, since \(x^2\) must be a multiple of 96, \(x\) must be a multiple of 24. Therefore, it is also a multiple of 12.
Sufficient

Stat (2)
Find the P.F. of 24.

> \(24 = 2^3 × 3^1\)

Bump up the exponents again.

> \(x^2\) is a multiple of \(2^4 × 3^2\)

> \(x\) is a multiple of \(2^2 × 3^1 = 12\)

Thus, \(x\) must be a multiple of 12.
Sufficient

(d) EACH statement ALONE is sufficient to answer the question asked.

Finally, Thank you! shoyland