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