Question of number properties

[KapilKapoor]

I got this question wrong in Veritas prep test paper; however i'm not satisfied with the solution and answer. The question is

Q. If n and y are different positive integers and n represents the number of different positive factors of y, is y a perfect square?
1. Root(n) is an odd integer
2. y=Root(5^(2(n-1)))

Ans given is A, i think answer should be D
What is others opinion ???

Regards,
Kapil

[Anurag@Gurome]

I got this question wrong in Veritas prep test paper; however i'm not satisfied with the solution and answer. The question is

Q. If n and y are different positive integers and n represents the number of different positive factors of y, is y a perfect square?
11. Root(n) is an odd integer
2. y=Root(5^(2(n-1)))

Ans given is A, i think answer should be D
What is others opinion ???

Regards,
Kapil

(1) √n is an odd integer implies that n must also be an odd integer.
Example: √49 = 7, √9 = 3, √121 = 11, but √100 = 10, √64 = 8
Also, all the powers of an odd integer are also odd, viz., 3² = 9, 3^3 = 27, 3^4 = 81 and so on.

It is given that n represents the number of different positive factors of y, which means that the question is asking if n is an odd integer. And it is clear that n is an odd integer; SUFFICIENT.

(2) y = √(5^(2(n-1)))
y = (5^(2(n-1)))^(1/2)
y = 5^(n - 1), but from this we cannot make out whether y is a perfect square or not; NOT sufficient.

The correct answer is A.

[KapilKapoor]

You are right, i don't know why but i had "n" as only odd values in my mind (may be because of stm 1) so i was thinking n-1=even => y = perfect square.
But i was wrong.
Thanks

[anuprajan5]

I got the answer A.

Statement 1 - My logic is that root n is an odd integer. Therefore n is an odd integer. (odd*odd = odd).
If there are odd number of factors and assuming y is a perfect square, it can work - For example - 49 -1,7,49 and 625 - 1,5,25,125,625
Taking any other square like 64 - 1,64,2,32,4,16,8,8. Therefore n will be even.

Statement 2 - y = 5 * root(5^(n-1)). y cannot be a perfect square.

Regards
Anup