Bunuel wrote:

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.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 NConsider 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)

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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 6Since 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 6Case b:

n = 6. Notice that 144/6² = 4, and 4 is an integer. In this case

n IS divisible by 6Since we cannot answer the

target question with certainty, statement 2 is NOT SUFFICIENT

Answer: A

Cheers,

Brent

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