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## For how many positive integer x is 130/x an integer?

tagged by: EconomistGMATTutor

This topic has 4 expert replies and 1 member reply

### Top Member

Vincen Legendary Member
Joined
07 Sep 2017
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550 messages
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#### For how many positive integer x is 130/x an integer?

Tue Oct 03, 2017 5:33 pm
For how many positive integer x is 130/x an integer?

(A) 8
(B) 7
(C) 6
(D) 5
(E) 3

The OA is A.

What is the fastest way to solve this PS question? Should I try number by number?

### GMAT/MBA Expert

Matt@VeritasPrep GMAT Instructor
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Wed Oct 11, 2017 5:40 pm
This is just a funky way of asking "How many distinct factors are there of 130?"

To see a nice way of doing this, check out this link!

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### GMAT/MBA Expert

Scott@TargetTestPrep GMAT Instructor
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Mon Oct 09, 2017 4:23 pm
Vincen wrote:
For how many positive integer x is 130/x an integer?

(A) 8
(B) 7
(C) 6
(D) 5
(E) 3
If x is a positive integer and 130/x is an integer, then x must be a factor of 130. Thus, we need to determine the number of factors of 130. To determine the number of factors, we break down 130 into primes and add 1 to the exponent of each unique prime, and then multiply. Thus:

130 = 10 x 13 = 2^1 x 5^1 x 13^1

Thus, 130 has (1 + 1)(1 + 1)(1 + 1) = 2 x 2 x 2 = 8 total factors.

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### GMAT/MBA Expert

Rich.C@EMPOWERgmat.com Elite Legendary Member
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Tue Oct 03, 2017 6:04 pm
Hi Vincen,

For 130/X to be an integer, X must be a factor of 130. There are a couple of different ways to determine all of the different factors of 130: you can 'brute force' the list or use prime-factorization.

If you prime-factor 130, you'll get...
(2)(65)
(2)(5)(13)

Thus, the factors of 130 are:
1
2
5
13
(2)(5) = 10
(2)(13) = 26
(5)(13) = 65
(2)(5)(13) = 130

Total = 8 (you should also notice that this is the largest possible answer among the 5 choices, so there's no way that you 'missed' any of them).

GMAT assassins aren't born, they're made,
Rich

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Contact Rich at Rich.C@empowergmat.com

### Top Member

Vincen Legendary Member
Joined
07 Sep 2017
Posted:
550 messages
6
Tue Oct 03, 2017 6:16 pm
Thanks Rich. It helps me. It was easier and faster than trying number by number.

### GMAT/MBA Expert

EconomistGMATTutor GMAT Instructor
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Fri Oct 06, 2017 11:42 am
Vincen wrote:
For how many positive integer x is 130/x an integer?

(A) 8
(B) 7
(C) 6
(D) 5
(E) 3

The OA is A.

What is the fastest way to solve this PS question? Should I try number by number?
Hi Vincen,
Let's take a look at your question.

for 130/x to be a positive integer, the x in the denominator should be one of the factors of 130. Let's factor out 130 using prime factorization.

130 = 2 x 5 x 13

All possible factors could be
1, 2, 5, 13, 130
and
2 x 5 = 10
2 x 13 = 26
5 x 13 = 65

So there are a total of 8 positive integers x so that 130/x is a positive integer.
1, 2, 5, 10, 13, 26, 65, 130

Therefore, Option A is correct.

I am available if you'd like any followup.

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