Sorrow wrote:

If n is a positive integer, and n^2 has 25 factors, which of the following must be true.

1. n has 12 factors.

2. n > 50

3. \(\sqrt n\) is an integer.

A. 1 and 2

B. 2 only

C. 3 only

D. 2 and 3

E. none

\(25=1 \times 25 = 5 \times 5\)

So prime factorization of \(n^2\) could be \(p^{24}\) or \(p^4q^4\) with \(p,q\) are distinct prime numbers.

Hence, \(n\) could be \(p^{12}\) or \(p^2q^2\)

(1) If \(n=p^{12}\) then \(n\) has \(12+1=13\) different factors, so (1) is not true.

(2) If \(n=p^2q^2\), for example \(n=2^2 \times 3^2 =36 <50\), so (2) is not true.

(3)

If \(n=p^{12} \implies \sqrt{n}=p^6\) is an integer.

If \(n=p^2q^2 \implies \sqrt{n}=pq\) is an integer.

So (3) is true.

The answer is C

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