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Re: is y a perfect square? [#permalink]
22 Aug 2011, 21:42

2

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Expert's post

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.

Re: is y a perfect square? [#permalink]
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

Re: is y a perfect square? [#permalink]
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.