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

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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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New post Updated on: 10 Jan 2016, 05:10
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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.
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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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New post 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.
Answer: I and III

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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New post 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.
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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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New post 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
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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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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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