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

Question

Is n an integer?

(1) 3n^2 is an integer.

(2)  \frac{\sqrt{n}}{3}  is an integer.

Archit3110 · Answer

#1
3n^2 is an integer.
n cna be integer or √n ; insufficient
#2
 \frac{\sqrt{n}}{3}  is an integer. 
n = 9,36,81,144 
n has to be an integer
sufficient
IMOB

freedom128 · Answer

Q.Is n an integer?

(1) 3n^2 is an integer. 
Pick a number, e.g.
- 3n^2 = 1, then n^2 = 1/3. n is NOT an integer. 
- 3n^2 = 3, then n^2 = 1. n is an integer. 
NOT SUFFICIENT

(2) √n/3 is an integer. 
- If √n is divisible by 3, n is definitely divisible by 9. Thus, n must be an integer. 
SUFFICIENT

Answer is (B) 
   Leave +1 kudo if you like the explanation

Bunuel · Answer

Official Explanation

We need to know whether n is an integer. We will most likely analyze by cases and by the rules of number properties. On to the data statements, separately first.

Statement (1) tells us that 3n^2 = integer. We can dig into some cases. Say 3n^2 = 18. That's an allowed case, since 18 is an integer. In this case n^2 = 6 and  n=\sqrt{6}  , so n is not an integer. We can probably find a case that generates a contradictory answer. Say 3n^2 = 27. That's an allowed case, since 27 is an integer. In this case, n = 3, so n is an integer. Hence we have contradictory answers from allowed cases, so we have insufficient information to answer the question definitively. Statement (1) is insufficient.

Statement (2) tells us that  \frac{\sqrt{n}}{3}=integer . That means that

\sqrt{n}=3*integer

\sqrt{n}=integer

This is similar to Statement (1), but different. Looking at the right side here: any integer times three yields another integer. So we are left with a (different) integer squared, and any integer squared yields another integer. That means that n will always be an integer. We have sufficient information to answer the question definitively, so Statement (2) is sufficient.

The correct answer is (B).

Bunuel · Answer

Video Explanation

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Cowtommi · Answer

What if n is 1/9 ?
which makes √n = 1/3, not an integer. but 1/3/3  is  is 9.

Bunuel · Answer

(1/3)/3 = 1/9, not 9.

Cowtommi · Answer

Thanks. What a shame of myself.

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

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

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