j_shreyans wrote:Is the positive integer n divisible by 6?

(1) n²/180 is an integer.

(2) 144/n² is an integer

OAA

NOTE: This question is VERY similar to this one: https://www.beatthegmat.com/ds-problem-t279217.html

Okay, 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)

For more on this concept, see the 2nd half of this free video: https://www.gmatprepnow.com/module/gmat- ... /video/825

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Okay, onto the question!

Target question: Is the positive integer n divisible by 6?

Statement 1: n²/180 is an integer
This tells us that n² is DIVISIBLE by 180
This means that 180 is "hiding in the prime factorization of n²
180 = (2)(2)(3)(3)(5)
So, n² = (2)(2)(3)(3)(5)(?)(?)(?)(?)

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

Rewrite as (n)(n) = [(2)(3)(5)(?)(?)][(2)(3)(5)(?)(?)]
This tells us that we can be certain that n = (2)(3)(5)(?)(?)
At this point it is clear that n is divisible by 6
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 6
Case b: n = 6. Notice that 144/6² = 4, and 4 is an integer. In this case n IS divisible by 6
Since we cannot answer the target question with certainty, statement 2 is NOT SUFFICIENT

Answer = A

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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