Is positive integer n divisible by 3 ?

Question

(1)  \frac{n^2}{36}  is an integer.

(2)  \frac{144}{n^2}  is an integer.

chix475ntu · Answer

st 1) N^2 / 36 is an integer
36 = 2^2 * 3^3
For N^2 to be evenly divided by 36, N should have factors at a minimum, one 2 and one 3. Sufficient
st 2) 144 / N^2 is an integer
144 = 2^4 * 3^2 
N could be either 2, 4, or 3. Not sufficient

A

TheGmatTutor · Answer

I agree with Chix47, with a little more detail...

S1. The prime factorization of 36 is 
2^2 * 3^2
therefore to be divisible by 36, both N^2 and N must have 3 as a factor
 S

S2. this statement is written differently from the first. notice that N^2 is in the denominator. Possible values for N^2 are 16, 9, 4. Therefore possible values for N are 4, 3, 2.

4/3 = not an integer
3/3 = an integer

therefore for S2 you get "sometimes yes, sometimes no" and it is insufficient. 
 Answer: A

Sash143 · Answer

From (1) n^2/36 = INT Therefore the values for n^2 that fit the condition = 36, 144 and so on n= 6, 12 (bcoz n is positive int) and so on. Fact (1) SUFF From (2) 144/n^2 = INT Therefore n^2 could be 1, 4, 9.... and when n= 1 (No), 2 (No), 3 (Yes). Fact 2 INSUFF Hence A :-D

stonecold · Answer

A very useful property -> X and X^n always have the exact same prime factors.
For N to be divisible by 3 => 3 must be the prime factors of N.

Statement 1=>
N^2/36 is an integer.
N^2 has both 2 and 3 as it prime.
Thus N must have 3 as its prime too.
Hence Sufficient.
Statement 2=>
Lets use test Cases here.
N=1 => 144/1^2 =Integer => 1 is not divisible by 3.
N=3 => 144/N^2 =integer => 3 is divisible by 3.
Hence not sufficient.

Hence A

anairamitch1804 · Answer

I did this problem with difference approach. Can some one please evaluate is my approach towards this problem correct?

Madhavi1990 · Answer

St 1: N sq/36 = integer. 36 is a multiple 3, so dvble by 3
St 2: 144/ N sq = integer. N = what? Not suff

Ans: A

agrasan · Answer

(1) We can write this statement as:
(n/6)^2 = integer

This implies that n is a multiple of 6, then it has to be a multiple of 3.
 Sufficient

(2) We can write this statement as:
(12/n)^2 = integer

This implies that 12 is a multiple of n but we can't say n will be a multiple of 3. Think of n = 2 and (12/2)^2 will be an integer.
 Insufficient

Hence,  (A)  is the answer.

Is positive integer n divisible by 3 ?

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