If n is a positive integer, is n^2 - n divisible by 12?

Question

(1) n/11 is an integer
(2) n/19 is an integer

M36-82

Bunuel · Accepted Answer

Official Solution:

If n is a positive integer, is  n^2 - n  divisible by 12? 
 
 (1)  \frac{n}{11}  is an integer 
 
The above means that  n  is a multiple of 11. 
 
If  n=11 , then  n^2 - n=n(n-1)=110=(not \ a \ multiple \ of \ 12); 
 
If  n=11*12 , then  n^2 - n=(a \ multiple \ of \ 12) - (a \ multiple \ of \ 12)=(a \ multiple \ of \ 12). 
 
Not sufficient.
 
(2)  \frac{n}{19}  is an integer 
 
The above means that  n  is a multiple of 19. 
 
If  n=19 , then  n^2 - n=n(n-1)=19*18=(not \ a \ multiple \ of \ 12); 
 
If  n=19*12 , then  n^2 - n=(a \ multiple \ of \ 12) - (a \ multiple \ of \ 12)=(a \ multiple \ of \ 12). 
 
Not sufficient.
 
(1)+(2)  n  is a multiple of both 11 and 19:
 
If  n=11*19 , then  n^2 - n=n(n-1)=11*19(11*19-1)=11*19*208=(not \ a \ multiple \ of \ 12); 
 
If  n=11*19*12 , then  n^2 - n=(a \ multiple \ of \ 12) - (a \ multiple \ of \ 12)=(a \ multiple \ of \ 12). 
 
Not sufficient.

Answer: E

chetan2u · Answer

n^2-n=n(n-1) , that is product of two consecutive integers.

We are looking for whether n*(n-1) has factor as 12.

Both combined tells us that n is divisible by 11 or 19, but nothing much on 12. But we should still go into more details, if we have time in actuals.

1) n/11 is an integer
Say n is 11*12, then n(n-1) is divisible by 12.
But say n is 11, then n(n-1)=11*10. Not divisible by 12.

2) n/19 is an integer
Say n is 19*12, then n(n-1) is divisible by 12.
But say n is 19, then n(n-1)=19*18. Not divisible by 12.

Combined 
Say n is 11*12*19, then n(n-1) is divisible by 12.
But say n is 11*19, then n(n-1)=11*19*(11*19-1)=11*19*208. Not divisible by 12.

E

TestPrepUnlimited · Answer

n^2 - n = n*(n - 1) , we want to check the factors for  n  and  n - 1 .

Statement 1:

We may write this as  n = 11*i  where i is any integer. We can only guarantee  n*(n - 1)  has a factor of 11, and there isn't much to say about  n - 1 = 11*i - 1 . Insufficient.

Statement 2:

Similar as above, we can only guarantee  n*(n - 1)  has a factor of 19.

Combined:

We know n is a multiple of 11*19 but this is still similar to what we had above (and we still know nothing useful about n - 1). Insufficient.

Ans: E

If n is a positive integer, is n^2 - n divisible by 12?

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 n^2 - n divisible by 12?

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