Is n an integer? (1) 2n is an integer. (2) 3n is an integer.

Question

Is n an integer?

(1) 2n is an integer. 
(2) 3n is an integer.

Bunuel · Answer

Is n an integer?

(1) 2n is an integer --> 2n=integer: if n=1/2 then the answer is NO but if n=1 then the answer is YES. Not sufficient.
(2) 3n is an integer --> 3n=integer: if n=1/3 then the answer is NO but if n=1 then the answer is YES. Not sufficient.

(1)+(2) Subtract (1) from (2): 3n-2n=integer-integer --> n=integer (since integer-integer=integer). Sufficient.

Answer: C.

Hope it's clear.

boomtangboy · Answer

+1 C

Bunuel like the shortcut for getting "n" while solving both. I was trying to use a fraction to break the question stem

ts2014 · Answer

thanks Bunuel!!!

i was taking into the realm of LCM's but couldnt get a headway. i.e., 1 and 2 cannot deduce the fact that n is an integer.

combining both of them together 2n and 3n is an integer, it would mean that 6n would be an integer but reached a block there again as it still doesnt prove n is an integer.

ur method is short and easy. thanks!!

VadimKlimenko · Answer

Is there any other way to solve the problem except "3n - 2n" approach?

Bunuel · Answer

Yes, it is. Though the approach provided is the easiest one.

From (1): 2n=a, for some integer a --> n=a/2.
From (2): 3n=b, for some integer b --> n=b/2.

a/2=b/3 --> a/b=2/3. Since a and b are integers, then a is a multiple of 2 --> a=2k, for some integer k --> 2n=2k --> n=k=integer. Sufficient.

Answer: C.

Hope it's clear.

VadimKlimenko · Answer

So for any integers a and b when a/b = k/m (a>k) (fractions simplifying) a is multiple of k, and b is multiple of b, isn't it ?
Delicious solution, thank you!

Bunuel · Answer

Yes, that's correct.

Bunuel · Answer

We have a/b=2/3. Since a and b are integers, then a is a multiple of 2 (or simply a must be an even number). Even number can be represented as a=2k, where k is an integer. If k=1/2, then a=2k=1=odd.

Hope it's clear.

VadimKlimenko · Answer

Hi, I had the same concern, let me answer your question:
We know that a/2=b/3 (if we had no info about a & b then a/2=b/3 could be anything, for example 1/3=0.333) and it gives us a/b = 2/3 (again, if no information about a & b it could be a = 1 and b = 1.5). Our key knowledge gives us that a = integer and b = integer and it means that minimum |a| = 2 and minimum |b|=3 and in general a is multiple of 2 and b is multiple of 3.

makhija1 · Answer

IMO the answer is C.

For statement 1, n=1/2 results in 2n = 1; not suff
For statement 2, n=1/3 results in 3n =1; not suff

Considering both the statements, 3n-2n=n; therefore n is an integer because the difference of two integer numbers is always integer. Suff.

ashwin1590 · Answer

What if the IInd statement says its 5N then also the Answer will be C ??

Bunuel · Answer

Yes.

2n = integer and 5n = integer gives 5n - 2n = 3n = integer. 2n = integer and 3n = integer gives 3n - 2n = n = integer.

Or: 2n = integer, means 2*2n = 4n = 2*integer = integer. So, if 5n = integer, then 5n - 4n = n = integer.

Hope it's clear.

sahilvijay · Answer

1) 2n is integer, 2n can be 1,2,3,4,5 which can give n as integer as well as fractions, A and D ruled out
2) 3n is integer same as above B ruled out.
combining both
if 2n is integer let say 2n=1 i.e. n=1/2 
and 3n is integer let say 3n=1 i.e. n=1/3
then the common series will be divisible by 6 (which is LCM of 2 and 3) i.e. 0,6,12,18,......and so on including negative values as well, clearly n is integer here.  Therefore Answer is C

Aamirso · Answer

1 may or may not be integer because of exceptions such as 1/2, 3/2 ...
2 may or may not be sufficient because of exceptions such as 1/3, 2/3...
Combining, both type of exceptions are eliminated and we know that N is an integer.

C is the answer.

Salsanousi · Answer

Statement 1)

2n = 1

Here n = 1/2 and no it is not an integer.

2n = 0

Here n = 0 and yes it is an integer.

Insufficient.

Statement 2)

In a similar way to statement 1 we can have n = 1/3 and n = 0 so insufficient.

Combined we know 2n = x and 3n = y

n = x/2

3x/2 = y

3x = 2y

Since they have no common factor we know that it must be an integer.

Posted from my mobile device

iamcabbage · Answer

In Bunuel's solution, why are we subtracting?

GMATinsight · Answer

iamcabbage 
The question is 
Is n an integer?

(1) 2n is an integer.
(2) 3n is an integer.

i.e we need to find property of n

When we take difference of two statements 
i.e. 3n - 2n = n  then we get n

also since 3n and 2n are both integer therefore,   3n-2n = Integer - Integer = Integer  , which also must be an Integer hence the two statements together are sifficient

Answer: Option C

iamcabbage · Answer

Hey, Thank you for responding. I am sorry but I am still confused. I mean, I understood the method, but I just wanted to understand the logic behind deciding to subtract. Why not add? I would really appreciate your help

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Is n an integer? (1) 2n is an integer. (2) 3n is an integer.

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