abhasjha wrote:Is positive integer n divisible by 3?

(1) n²/36 is an integer
(2) 144/n² is an integer

A lot of integer property questions can be solved using prime factorization.
For questions involving divisibility, divisors, factors and multiples, we can say:
If N is divisible by k, then k is "hiding" within the prime factorization of N

Consider these examples:
24 is divisible by 3 because 24 = (2)(2)(2)(3)
Likewise, 70 is divisible by 5 because 70 = (2)(5)(7)
And 112 is divisible by 8 because 112 = (2)(2)(2)(2)(7)
And 630 is divisible by 15 because 630 = (2)(3)(3)(5)(7)
------------------------------

Okay, onto the question!

Target question: Is positive integer n divisible by 3?

Statement 1: n²/36 is an integer
This tells us that n² is DIVISIBLE by 36
This means that 36 is "hiding in the prime factorization of n²
36 = (2)(2)(3)(3)
So, n² = (2)(2)(3)(3)(?)(?)(?)(?)

Aside: the (?)'s represent other possible primes in the prime factorization of n²

Rewrite as (n)(n) = [(2)(3)(?)(?)][(2)(3)(?)(?)]
This tells us that we can be certain that n = (2)(3)(?)(?)
At this point it is clear that n is divisible by 3
Since we can answer the target question with certainty, statement 1 is SUFFICIENT

Statement 2: 144/n² is an integer
There are several values of n that satisfy this condition. Here are two:
Case a: n = 1. Notice that 144/1² = 144, and 144 is an integer. In this case n is NOT divisible by 3
Case b: n = 6. Notice that 144/6² = 4, and 4 is an integer. In this case n IS divisible by 3
Since we cannot answer the target question with certainty, statement 2 is NOT SUFFICIENT

Answer = A

--------------------

Here are a few more questions that require us to understand the relationship between divisibility and prime factorization:
https://www.beatthegmat.com/factors-t272765.html
https://www.beatthegmat.com/do-t-and-12- ... 74544.html
https://www.beatthegmat.com/is-xy-a-mult ... 74522.html
https://www.beatthegmat.com/confused-nee ... 71655.html
https://www.beatthegmat.com/multiple-of-990-t272719.html
https://www.beatthegmat.com/if-n-t-3-for ... 48420.html

Cheers,
Brent

Gmat_Geek wrote:I have tried like this:
for positive value of n:
Is n/3=?
or n=?

S1: n^2/36=x
n=6*root(x)

In this case we are not sure of value x, ie we can have any value for x for n. hence INSUFFICIENT !

Please let me know, if its correct approach.

In your approach, it's important to note that x is a perfect square.

If j and k are integers, and j²/k² is an integer, then j²/k² is a perfect square.

So, for the question, n²/36 is an integer.
In your solution, you state that n²/36 = x, which means n = 6√x
However, since n²/6² (aka n²/36) is an integer, we know (from the above rule) that n²/6² (aka x) is a perfect square.
So, √x is an integer, which means 6√x is definitely divisible by 3.

Cheers,
Brent