WishMasterUA wrote:

if n and y are positive integers and n represents the number of different positive factors of y, is y a perfect square?

1) \(\sqrt{n}\) is an odd number.

2) \(y=\sqrt{5^{2(n-1)}}\)

The question is based on the following concept:

If a number has odd number of factors, it must be a perfect square.

If a number has even number of factors, it cannot be a perfect square.

For why and how, check:

http://www.veritasprep.com/blog/2010/12 ... t-squares/n is the number of positive factors of y.

Question: Is y a perfect square?

Re-state the question as: Is n an odd integer?

Statement 1: \(\sqrt{n}\) is an odd number.

If \(\sqrt{n}\) is an odd number, n must be an odd number.

(All powers of an odd number are odd. If a is odd, a^2 is odd, a^3 is odd, a^4 is odd, \(\sqrt{n}\), if integral, is odd etc.)

Since we know that n is odd, this statement is sufficient.

Statement 2: \(y=\sqrt{5^{2(n-1)}}\)

\(y=5^{n-1}\)

Obviously, the number of factors of y is n. Do we know whether n is odd? No, we don't.

Hence this statement alone is not sufficient.

Answer (A)

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