If n is a positive integer, is 169 a factor of n?

Question

(1) 52 is the greatest common divisor of 260 and n
(2) 1,352 is the least common multiple of 104 and n

CareerGeek · Accepted Answer

If n is a positive integer, is 169 a factor of n?

(1) 52 is the greatest common divisor of 260 and n
260 = 52*5
Possible values of 
n = 52 —> 169 is not a divisor of n
n = 52*13 —> 169 is a divisor of n

Insufficient

(2) 1,352 is the least common multiple of 104 and n

1352 = 169*8 =  13^2 *8
104 = 13*8
Possible values of 
n =  13^2  —> 169 is a factor of n
n =  13^2 *2 —> 169 is a factor of n
n =  13^2 *4 —> 169 is a factor of n
n =  13^2 *8 —> 169 is a factor of n

Sufficient

IMO Option B

Pls Hit kudos if you like the solution

Posted from my mobile device

manass · Answer

(1) 52 is the greatest common divisor of 260 and n
 not sufficient

(2) 1,352 is the least common multiple of 104 and n
divide number by 104 and 169 you can tell the number is divisible by 169
sufficient

hiranmay · Answer

If n is a positive integer, is 169 a factor of n?
 Given: n > 0 & integer
n = 169*m ? where m is an integer from 1 to any m
=> n = 13^2*m?

(1) 52 is the greatest common divisor of 260 and n -->  insufficient : 260 = 52*5, so GCD of 52*5 and n = 52, so n = 52*p=13*4*p, but p can or can't be multiple of 13
(2) 1,352 is the least common multiple of 104 and n -->  sufficient : 104 = 13*2^3, so LCM of 13*2^3 & n = 1,352 = 13^2*2^3, so n must be multiple of 13^2

So the  answer is B

Kinshook · Answer

If n is a positive integer, is 169 a factor of n?

(1) 52 is the greatest common divisor of 260 and n
52=2*2*13
n= 2^2*13k 
But 169= 13^2 
and n is not necessarily a multiple of 169
or 169 is not necessarily a factor of n
NOT SUFFICIENT

(2) 1,352 is the least common multiple of 104 and n
1352=13*104 =  2^3*13^2 
104=2*2*2*13 =  2^3*13 
Since 1352 is least common multiple of 104 and n
=> 13^2  is a factor of n
=> 169 is a factor of n
SUFFICIENT

IMO B

Abhi077 · Answer

If n is a positive integer, is 169 a factor of n?

(1) 52 is the greatest common divisor of 260 and n
(2) 1,352 is the least common multiple of 104 and n

Question Stem Analysis:

According to the stem, we must find out whether n/169 , i.e  \frac{n}{13^2}  is an integer.

Statement One Alone:

Prime factorization of 52 is 13 X  2^2 
Prime factorization of 260 is 13 X 5 X  2^2

The greatest common divisor (gcd), also known as the greatest common factor (gcf), or highest common factor (hcf), of two or more non-zero integers, is the largest positive integer that divides the numbers without a remainder.

For GCD, we pick the lowest power of common factors. we know that n consist a 13, but n can or cannot contain more than  13^2 , because gcf of 260 and n is limited to 13 which we have in 260.

Hence this statement alone is not sufficient.

Statement Two Alone:

Prime factorization of 104 is 13 X 2^3 
Prime factorization of 1352 is 13 X 13 X  2^3

The lowest common multiple or lowest common multiple (lcm) or smallest common multiple of two integers a and b is the smallest positive integer that is a multiple both of a and of b.

For LCM, we pick the highest power of common factors. we know that 104 just has  13^1 , and that 1352 has  13^2  which definitively must have come from n. Hence we can say that n has 13^2  and that will make it divisible by 169.

Statement two alone is sufficient.

Hence the answer must be  B

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 n is a positive integer, is 169 a factor of n?

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 Data Sufficiency (DS) Questions

If n is a positive integer, is 169 a factor of n?

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