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If y is a positive integer, is (y^3 + 5)^2/4 an integer?

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If y is a positive integer, is \(\frac{(y^3 + 5)^2}{4}\) an integer?

1) The square root of y has three prime factors.
2) Each prime factor of \(y^3\) is greater than 5.
[Reveal] Spoiler: OA

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If y is a positive integer, is (y^3 + 5)^2/4 an integer? [#permalink]

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New post 30 Oct 2017, 09:04
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sasyaharry wrote:
If y is a positive integer, is \(\frac{(y^3 + 5)^2}{4}\) an integer?

1) The square root of y has three prime factors.
2) Each prime factor of \(y^3\) is greater than 5.



hi..

\(\frac{(y^3 + 5)^2}{4}\) will be an integer if y is ODD as \(y^3+5\) will become even and its SQUARE will be div by 4..

lets see the statements

1) The square root of y has three prime factors.
If one prime factor is 2, ans is NO as \(y^3+5\) will be ODD
if all 3 prime factors are ODD, ans is YES
insuff

2) Each prime factor of \(y^3\) is greater than 5.
MEANS all prime factor are ODD, so \(y^3+5\) will be EVEN
ans is YES
suff

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Combination of similar and dissimilar things : http://gmatclub.com/forum/topic215915.html

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If y is a positive integer, is (y^3 + 5)^2/4 an integer? [#permalink]

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New post 31 Oct 2017, 09:28
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Some folks might find this helpful.

\(\frac{(y^3 + 5)^2}{4}\) = Some integer , lets say K.
\((y^3 + 5)^2 = 4K\) ; RHS is EVEN since 4K is also a multiple of 2.
\((y^3 + 5)^2\) = EVEN
\((y^3 + 5)^2\) can be EVEN only if \((y^3 + 5)\) is EVEN.
\((y^3 + 5)\) can be even only if \(y^3\) is ODD
\(y^3\) is ODD only if y is ODD.

Hence the question boils down to, is Y an odd integer?

Statement 1: Not sufficient for reasons described in the post above.

Statement 2: Each prime factor of \(y^3\) is ODD

A handy rule to remember is that \(N\) and \(N^x\) have the same prime factors.

Hence, if \(Y^3\) has odd prime factors, \(Y\) also has odd prime factors.
Since the only even prime factor is 2. The number Y is odd. Sufficient.
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If y is a positive integer, is (y^3 + 5)^2/4 an integer?   [#permalink] 31 Oct 2017, 09:28
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