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.
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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