If x and y are positive integers and 180x=y^3, which of the following

Question

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

Bunuel · Accepted Answer

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.

gracie · Answer

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

ScottTargetTestPrep · Answer

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

BlueRiceCakes · Answer

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?

Bunuel · Answer

bumpbot · Answer

Automated notice from GMAT Club BumpBot:

A member just gave Kudos to this thread, showing it’s still useful. I’ve bumped it to the top so more people can benefit. Feel free to add your own questions or solutions.

This post was generated automatically.

If x and y are positive integers and 180x=y^3, which of the following

Prep Toolkit

Top 5 GMAT Debrief Videos

615 to 715 in 15 Days (Ria's Strategies)

More than Quant/Verbal Abilities: Speed vs. Accuracy

745 with Only Self-Prep and GMAT Club

From 555 to 765: 210-point Improvement

Perfect 805 Score Debrief by Julia

My Rewards

GMAT Problem Solving (PS) Questions

If x and y are positive integers and 180x=y^3, which of the following

Prep Toolkit

615 to 715 in 15 Days (Ria's Strategies)

More than Quant/Verbal Abilities: Speed vs. Accuracy

745 with Only Self-Prep and GMAT Club

From 555 to 765: 210-point Improvement

Perfect 805 Score Debrief by Julia

My Rewards