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12 Nov 2014, 10:05
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A
Be sure to select an answer first to save it in the Error Log before revealing the correct answer (OA)!
Difficulty:
55%
(hard)
Question Stats:
62% (01:58) correct
38%
(02:03)
wrong
based on 380
sessions
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Not Attempted Yet
Tough and Tricky questions: Divisibility.
Is the positive integer \(n\) divisible by \(6\)?
(1) \(\frac{n^2}{180}\) is an integer.
(2) \(\frac{144}{n^2}\) is an integer.
Kudos for a correct solution.
20 Apr 2018, 07:54
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Bunuel
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 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
Cheers,
Brent
General Discussion
12 Nov 2014, 21:27
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Statement 1: n^2/180 is an integer.
n^2 = 180*a = 6^2*5 *a [Where a is an integer]
Therefore n has to be a multiple of 6. Sufficient
Statement 2 : 144/n^2 is an integer.
n = 1, when n is not divisible by 6
n= 12, when n is divisible by 6
Therefore, insufficient
Therefore the answer has to be A)
n^2 = 180*a = 6^2*5 *a [Where a is an integer]
Therefore n has to be a multiple of 6. Sufficient
Statement 2 : 144/n^2 is an integer.
n = 1, when n is not divisible by 6
n= 12, when n is divisible by 6
Therefore, insufficient
Therefore the answer has to be A)
