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# If 375y = x^2, and x and y are positive integers then which of the fol

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Joined: 18 Sep 2014
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If 375y = x^2, and x and y are positive integers then which of the fol  [#permalink]

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Updated on: 10 Jan 2016, 05:10
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Difficulty:

45% (medium)

Question Stats:

69% (01:43) correct 31% (02:18) wrong based on 145 sessions

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If $$375y = x^2$$, and $$x$$ and $$y$$ are positive integers then which of the following must be an integer?

I. $$\frac{y}{15}$$

II. $$\frac{y}{30}$$

III. $$\frac{(y^2)}{25}$$

a) I only
b) III only
c) I and II
d) I and III
e) I, II and III

Originally posted by Nevernevergiveup on 08 Jan 2016, 11:26.
Last edited by Bunuel on 10 Jan 2016, 05:10, edited 1 time in total.
e
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Joined: 11 Sep 2015
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Re: If 375y = x^2, and x and y are positive integers then which of the fol  [#permalink]

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08 Jan 2016, 11:35
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Nevernevergiveup wrote:
If $$375y = x^2$$, and $$x$$ and $$y$$ are positive integers then which of the following must be an integer?

I. $$\frac{y}{15}$$
II. $$\frac{y}{30}$$
III. $$\frac{(y^2)}{25}$$

375 = (3)(5)(5)(5) = (3)(5)(5²)
In order for 375y to be a perfect square, the prime factorization of y must contain at least one 3 and one 5.
In other words, y must be a multiple of 15.

If y is a multiple of 15, then y/15 must be an integer and y²/25 must be an integer.

Here's a free video on the squares of integers: http://www.gmatprepnow.com/module/gmat- ... /video/829
It pertains to this question.

Cheers,
Brent
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Re: If 375y = x^2, and x and y are positive integers then which of the fol  [#permalink]

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08 Jan 2016, 12:06
Nevernevergiveup wrote:
If $$375y = x^2$$, and $$x$$ and $$y$$ are positive integers then which of the following must be an integer?

I. $$\frac{y}{15}$$
II. $$\frac{y}{30}$$
III. $$\frac{(y^2)}{25}$$

a) I only
b) III only
c) I and II
d) I and III
e) I, II and III

Given, $$375y = x^2$$ ---> as x and y are both integers ---> 375*y must become a perfect square for x to be an integer

375*y ---> $$3*5^3*y$$

Thus, from the given information, $$x^2=3*5^3*y$$ ---> y MUST be = $$p*3^{odd}*5^{odd}$$, where p = integer. Possible values of y = $$3*5$$ or $$3^3*5$$ or $$3^3*5^3$$ etc and in all cases you can see that y = 15p or y is a multiple of p.

Going back to the options,

I) y/15 MUST be an integer as shown above. Must be a true statement.
II) y/30, as we see that y = 15p but it may or may not be even multiple of 15. y = 15 or 75 are equally applicable to the conditions given but is not divisible by 30. Hence this statement is not a must be true statement.
III) y^2/25 is an integer . As y = 15p ---> y^2 = 225p = 25*9p = 25*q. Thus y^2 is a multiple of 25 and hence this statement is a must be true.

D is thus the correct answer.
VP
Joined: 07 Dec 2014
Posts: 1114
If 375y = x^2, and x and y are positive integers then which of the fol  [#permalink]

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26 May 2017, 19:26
Nevernevergiveup wrote:
If $$375y = x^2$$, and $$x$$ and $$y$$ are positive integers then which of the following must be an integer?

I. $$\frac{y}{15}$$

II. $$\frac{y}{30}$$

III. $$\frac{(y^2)}{25}$$

a) I only
b) III only
c) I and II
d) I and III
e) I, II and III

based on answer choices, assume y has a ones unit of 5
try 5*375 no
try 15*375 yes=75^2
y=15
I and III
D
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Joined: 09 Sep 2013
Posts: 8789
Re: If 375y = x^2, and x and y are positive integers then which of the fol  [#permalink]

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07 Sep 2018, 13:01
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Re: If 375y = x^2, and x and y are positive integers then which of the fol &nbs [#permalink] 07 Sep 2018, 13:01
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