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If n and y are positive integers and n represents the number

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If n and y are positive integers and n represents the number [#permalink] New post 22 Aug 2011, 05:34
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42% (02:15) correct 58% (01:44) wrong based on 26 sessions
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 odd number

(2) \(y=\sqrt{5^{2(n-1)}}\)
[Reveal] Spoiler: OA

Last edited by Bunuel on 11 Jul 2013, 00:56, edited 3 times in total.
Edited the question.
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Re: is y a perfect square? [#permalink] New post 22 Aug 2011, 09:00
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) square(n) is odd number

2) y=square(5^(2(n-1))


I think 'square' means 'square-root'. Considering it true, my explanation


\sqrt{n} is an odd number => n would also be an odd number
=> Sufficient to determine, Y would not have 'even number of factors' and that means it can't a perfect square.

y = \sqrt{5^(2(n-1))} => y= 5^(n-1) => if n is odd than (n-1) would be a perfect square but if n is even, y would not be perfect answer.
=> NOT Sufficient

So answer is A
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Re: is y a perfect square? [#permalink] New post 22 Aug 2011, 09:53
anordinaryguy wrote:
=> Sufficient to determine, Y would not have 'even number of factors' and that means it can't a perfect square.


Can you explain that a bit more?

Edit: Found an answer myself: mathforum. org/library/drmath/view/72126.html
Such specific information...
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Re: is y a perfect square? [#permalink] New post 22 Aug 2011, 10:20
anordinaryguy wrote:
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) square(n) is odd number

2) y=square(5^(2(n-1))


I think 'square' means 'square-root'. Considering it true, my explanation


\sqrt{n} is an odd number => n would also be an odd number
=> Sufficient to determine, Y would not have 'even number of factors' and that means it can't a perfect square.

y = \sqrt{5^(2(n-1))} => y= 5^(n-1) => if n is odd than (n-1) would be a perfect square but if n is even, y would not be perfect answer.
=> NOT Sufficient

So answer is A



For (1) my understanding is different. A perfect square must have event exponents on it's prime factorization, but an odd number of factors. So like you said, root n is odd, meaning Y has an odd number of factors. If a number has an odd number of factors it is a perfect square.

e.g.

1X4
2X2

3 factors

1x64
2x32
4x16
8x8

7 factors


The perfect square always has odd factors.
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Re: is y a perfect square? [#permalink] New post 22 Aug 2011, 15:27
if a number equals = (p1^k)* (p2^l)*(P3^m)....
total no of factors= (k+1)*(l+1)*(m+1)....

If Sqrtroot of n is add => n is also odd.
here n= (k+1)*(l+1)*(m+1)....

now for n to be odd each of k, l,m should be even which implies that y is a perfect square.
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Re: is y a perfect square? [#permalink] New post 22 Aug 2011, 15:28
in the above post, p1, p2, p3 are prime factors.
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Re: is y a perfect square? [#permalink] New post 22 Aug 2011, 21:42
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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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Re: is y a perfect square?   [#permalink] 22 Aug 2011, 21:42
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