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

Can you explain that a bit more?

Edit: Found an answer myself: mathforum. org/library/drmath/view/72126.html
Such specific information...

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.

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.

in the above post, p1, p2, p3 are prime factors.

Official solution from Veritas Prep.

Correct Answer: A

This difficult question depends on an understanding of the properties of factors of a perfect square. The only way that a number can have an odd number of different total factors is if that number is a perfect square. Consider three numbers that are perfect squares, all of which have an odd number of total factors: 1, 9, and 16. The number 1 has only one unique factor (1); the number 9 has exactly three different factors (1, 3, 9); and 16 has five total factors (1, 16, 2, 8, 4). Statement (1) proves that n must be an odd integer so the information is sufficient to prove that y has an odd number of total unique factors and thus must be a perfect square. Statement (2) is more difficult to assess and requires that several different possible values of n be considered. For instance if n = 1 then y would be equal to 1 and it would indeed be a perfect square. However, if n = 2 then y would be equal to 5 which is not a perfect square. From those two values of n, it is clear that statement (2) is not sufficient and the correct answer is A, statement 1 alone is sufficient.

Statement 1 says square root of n is odd -> n is also odd and we know that total number of factors of any number can be derived using (n+1)(m+1)... where n & m are the power of any prime factors of a number. (Example: 108 = 2^2*3^3, so the total number of factors is (2+1)*(3+1) = 3*4 = 12 ) Hence, for perfect number such as 4, 9, total number of factors will always be an odd, unlike non perfect numbers, since (even + 1 = odd) -> (2+1 = 3, 6+1 =7). Hence, since statement 1 clearly says y has an odd number of factors, y is definitely a perfect square. Sufficient!

With Statement 2, we can easily put values of n as 1, 2 & 3 and then we can easily derive that with n=2, y is 5 which is not a perfect square and with n=3, y=25 -> clearly a perfect square. Hence, Not sufficient!

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