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

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Senior Manager
Joined: 22 Nov 2016
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If y is a positive integer, is (y^3 + 5)^2/4 an integer? [#permalink]

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30 Oct 2017, 08:20
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85% (hard)

Question Stats:

34% (01:48) correct 66% (02:04) wrong based on 35 sessions

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

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30 Oct 2017, 09:04
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Expert's post
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

B
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Absolute modulus :http://gmatclub.com/forum/absolute-modulus-a-better-understanding-210849.html#p1622372
Combination of similar and dissimilar things : http://gmatclub.com/forum/topic215915.html

Kudos [?]: 5860 [2], given: 117

Senior Manager
Joined: 22 Nov 2016
Posts: 251

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

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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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