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rraggio
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If x and y are positive integers and 180x=y^3, which of the following 04 Sep 2010, 11:34
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76% (01:23) correct 24% (02:02) wrong based on 1813 sessions

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If x and y are positive integers and 180x = y^3, which of the following must be an integer?

I. \(\frac{x}{(2^2*3*5)}\)

II. \(\frac{x}{(2*3^2*5)}\)

III. \(\frac{x}{(2*3*5^2)}\)


(A) None
(B) I only
(C) II only
(D) III only
(E) I and II
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D
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Bunuel
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If x and y are positive integers and 180x=y^3, which of the following 04 Sep 2010, 12:01
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rraggio
If x and y are positive integers and 180x=y^3, which of the following must be an integer?

I. x/(2^2*3*5)
II. x/(2*3^2*5)
III. x/(2*3*5^2)

(A) None
(B) I only
(C) II only
(D) III only
(E) I and II
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\(x\) and \(y\) are integers and \(180x=y^3\) --> \(180x\) equals to cube of an integer (\(y^3\)).

\(180x=2^2*3^2*5*x=y^3\). The smallest value of \(x\) for which \(2^2*3^2*5*x\) is a cube of an integer is when \(x=2*3*5^2\) (to complete powers of 2, 3 and 5 to cubes). In this case \(180x=(2^2*3^2*5)*(2*3*5^2)=(2*3*5)^3=cube \ of \ an \ integer\). So \(x\) is definitely divisible by \(2*3*5^2\).

Answer: D.

Similar problems:
https://gmatclub.com/forum/number-prope ... ne#p712584
https://gmatclub.com/forum/og-quantitat ... ml#p704028
https://gmatclub.com/forum/division-fac ... ml#p666722

Hope it helps.­
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If x and y are positive integers and 180x=y^3, which of the following 15 May 2018, 10:46
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rraggio
If x and y are positive integers and 180x = y^3, which of the following must be an integer?

I. \(\frac{x}{(2^2*3*5)}\)

II. \(\frac{x}{(2*3^2*5)}\)

III. \(\frac{x}{(2*3*5^2)}\)


(A) None
(B) I only
(C) II only
(D) III only
(E) I and II
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180=5*6*6
let x=5*5*6 to produce (5*6)^3=y^3
5*5*6=150
only III is an integer
D
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If x and y are positive integers and 180x=y^3, which of the following 31 May 2020, 11:20
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rraggio
If x and y are positive integers and 180x = y^3, which of the following must be an integer?

I. \(\frac{x}{(2^2*3*5)}\)

II. \(\frac{x}{(2*3^2*5)}\)

III. \(\frac{x}{(2*3*5^2)}\)


(A) None
(B) I only
(C) II only
(D) III only
(E) I and II
Show more

Solution:

Since 180x = y^3, we see that 180x must be a perfect cube. Recall that a perfect cube has prime factors whose exponents are positive multiples of 3. Let’s first factor 180:

180 = 18 x 10 = 2^2 x 3^2 x 5^1

Now, to make the expression 180x a perfect cube, we need x to contain at least one 2, at least one 3, and at least two fives. (If that happened, then 180x would be 2^3 x 3^3 x 5^3, which would make it a perfect cube.)

Since we are given that 180x is a perfect cube, the minimum value of x is 2^1 x 3^1 x 5^2, so only III MUST be an integer.

Answer: D
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If x and y are positive integers and 180x=y^3, which of the following 14 Aug 2024, 17:30
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If x and y are positive integers and 180x = y^3, which of the following must be an integer?

I. x(22∗3∗5)x(22∗3∗5)

II. x(2∗32∗5)x(2∗32∗5)

III. x(2∗3∗52)x(2∗3∗52)


(A) None
(B) I only
(C) II only
(D) III only
(E) I and II
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I'm a noob so help me out

Considering x = 150 (since 150x180 = 27000 which is 30^3)
x = 150 only satisfies III

Considering x = 1200, (since 180x1200 = 216000 which is 60^3)
x = 1200 satisfies options options I and III both.

Am I doing something wrong?
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If x and y are positive integers and 180x=y^3, which of the following 15 Aug 2024, 00:16
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BlueRiceCakes

Quote:
If x and y are positive integers and 180x = y^3, which of the following must be an integer?

I. \(\frac{x}{(2^2*3*5)}\)

II. \(\frac{x}{(2*3^2*5)}\)

III. \(\frac{x}{(2*3*5^2)}\)

(A) None
(B) I only
(C) II only
(D) III only
(E) I and II­
I'm a noob so help me out

Considering x = 150 (since 150x180 = 27000 which is 30^3)
x = 150 only satisfies III

Considering x = 1200, (since 180x1200 = 216000 which is 60^3)
x = 1200 satisfies options options I and III both.

Am I doing something wrong?
Show more
­
The question asks which of the following MUST be an integer, not COULD be an integer. While all options COULD result in an integer for SOME particular values of x, only option D will be an integer for ALL valid values of x. To determine which MUST be an integer, we consider the least value of x possible.

Check similar questions to practice:
https://gmatclub.com/forum/what-is-the- ... 21070.html (GMAT Club Tests)
https://gmatclub.com/forum/what-is-the- ... 68984.html (GMAT Club Tests)
https://gmatclub.com/forum/if-n-and-y-a ... 92562.html
https://gmatclub.com/forum/if-n-is-a-po ... 29929.html
https://gmatclub.com/forum/if-p-is-a-po ... 98638.html
https://gmatclub.com/forum/if-x-is-a-po ... 88388.html
https://gmatclub.com/forum/x-is-divisib ... 06841.html
https://gmatclub.com/forum/if-x-and-y-a ... 30696.html
https://gmatclub.com/forum/if-5400mn-k- ... 09284.html
https://gmatclub.com/forum/if-m-and-n-a ... 08985.html
https://gmatclub.com/forum/if-n-and-y-a ... 92562.html

Hope it helps.­
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