It is currently 23 Feb 2018, 04:54

GMAT Club Daily Prep

Thank you for using the timer - this advanced tool can estimate your performance and suggest more practice questions. We have subscribed you to Daily Prep Questions via email.

Customized
for You

we will pick new questions that match your level based on your Timer History

Track

every week, we’ll send you an estimated GMAT score based on your performance

Practice
Pays

we will pick new questions that match your level based on your Timer History

Events & Promotions

Events & Promotions in June
Open Detailed Calendar

If x and y are positive integers, is the total number of positive divi

Author Message
TAGS:

Hide Tags

Intern
Joined: 06 Oct 2016
Posts: 4
If x and y are positive integers, is the total number of positive divi [#permalink]

Show Tags

25 Dec 2016, 05:23
6
This post was
BOOKMARKED
00:00

Difficulty:

95% (hard)

Question Stats:

32% (01:39) correct 68% (01:20) wrong based on 72 sessions

HideShow timer Statistics

If x and y are positive integers, is the total number of positive divisors of x^3 a multiple of the total number of positive divisors of y^2?

(1) x = 4

(2) y = 6
[Reveal] Spoiler: OA

Last edited by vienbuuchau on 25 Dec 2016, 05:38, edited 2 times in total.
Math Expert
Joined: 02 Sep 2009
Posts: 43894
Re: If x and y are positive integers, is the total number of positive divi [#permalink]

Show Tags

25 Dec 2016, 05:33
vienbuuchau wrote:
If x and y are positive integers, is the total number of positive divisors of x^3 a multiple of the total number of positive divisors of y^2?

(1) x = 4

(2) y = 6

_________________
Math Expert
Joined: 02 Sep 2009
Posts: 43894
Re: If x and y are positive integers, is the total number of positive divi [#permalink]

Show Tags

25 Dec 2016, 05:34
3
KUDOS
Expert's post
5
This post was
BOOKMARKED
vienbuuchau wrote:
If x and y are positive integers, is the total number of positive divisors of x^3 a multiple of the total number of positive divisors of y^2?

(1) x = 4

(2) y = 6

Finding the Number of Factors of an Integer

First make prime factorization of an integer $$n=a^p*b^q*c^r$$, where $$a$$, $$b$$, and $$c$$ are prime factors of $$n$$ and $$p$$, $$q$$, and $$r$$ are their powers.

The number of factors of $$n$$ will be expressed by the formula $$(p+1)(q+1)(r+1)$$. NOTE: this will include 1 and n itself.

Example: Finding the number of all factors of 450: $$450=2^1*3^2*5^2$$

Total number of factors of 450 including 1 and 450 itself is $$(1+1)*(2+1)*(2+1)=2*3*3=18$$ factors.

If x and y are positive integers, is the total number of positive divisors of x^3 a multiple of the total number of positive divisors of y^2?

(1) $$x=4$$. From this statement we have that$$x^3=64=2^6$$, thus the number of factors of 64 is 6+1=7.

Now, may $$y^2$$ have the number of factors which is factor of 7, so 1 or 7 factors? Well may have or may not. Number of factors of a perfect square is odd. So $$y^2$$ should have either 1 factor (for example if y^2=1^2) or 7 (for example if y^2=81^2=3^6 or y^2=8^2=2^6), both are possible, BUT $$y^2$$ can have other odd number of factors say 3 (for example if y=5^2) and 3 is not factor of 7. Not sufficient

(2) $$y=6$$. From that: $$y^2=36=2^2*3^2$$, thus the number of factors of 36 is (2+1)*(2+1)=9.

Can $$x^3$$ have the number of factors which is multiple of 9 (9, 18, 27, ...)? Let's represent $$x$$ as the product of its prime factors: $$x^3=(a^p*b^q*c^r)^3=a^{3p}*b^{3q}*c^{3r}$$. The number of factors would be $$(3p+1)(3q+1)(3r+1)$$ and this should be multiple of 9. BUT $$(3p+1)(3q+1)(3r+1)$$ is not divisible by 3, hence it can not be multiple of 9. The answer is NO. Sufficient.

Not to complicate $$x^3$$ has $$3k+1&gt;$$ number of distinct factors (1, 4, 7, 10, ... odd or even number), so the number of factors of $$x^3$$ is 1 more than a multiple of 3, thus it's not divisible by 3 and hence not by 9.

_________________
Manager
Joined: 01 Sep 2016
Posts: 207
GMAT 1: 690 Q49 V35
Re: If x and y are positive integers, is the total number of positive divi [#permalink]

Show Tags

18 Sep 2017, 23:36
AWESOME, I saw a very similar question somewhere else. Hats off to Bunuel for such a wonderful explanation. Bunuel: you were right. they are very similar. I made a note. thanks
_________________

we shall fight on the beaches,
we shall fight on the landing grounds,
we shall fight in the fields and in the streets,
we shall fight in the hills;
we shall never surrender!

Re: If x and y are positive integers, is the total number of positive divi   [#permalink] 18 Sep 2017, 23:36
Display posts from previous: Sort by