If x and y are Positive Integers, is x^16−y^8+345y^2 Divisible by 15? GMAT Data Sufficiency

Question: If x and y are positive integers, \(x^{16}-y^8+345y^2\) is divisible by 15?

1. x is a multiple of 25, and y is a multiple of 20
2. \(y=x^2\)

1. Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient.
2. Statement (2) ALONE is sufficient, but statement (1) alone is not sufficient.
3. BOTH statements TOGETHER are sufficient, but NEITHER statement ALONE is sufficient.
4. EACH statement ALONE is sufficient.
5. Statements (1) and (2) TOGETHER are NOT sufficient.

“If x and y are positive integers, is \(x^{16}-y^8+345y^2\) divisible by 15?”- is a topic of the GMAT Quantitative reasoning section of GMAT. This question has been taken from the book "GMAT Quantitative Review". GMAT Quant section consists of a total of 31 questions. GMAT Data Sufficiency questions consist of a problem statement followed by two factual statements. GMAT data sufficiency comprises 15 questions which are two-fifths of the total 31 GMAT quant questions.

Answer:

Approach Solution (1):

First of all notice that since \(345y^2\) is divisible by 15, we can drop it (this term won’t affect the remainder)

(1) x is a multiple of 25, and y is a multiple of 20. Now both x and y could be multiples of 15 as well (eg. x = 25 * 15 and y = 20 * 15) and in this case \(x^{16}-y^8=15*(...)\) will be divisible by 15 OR one could be multiple of 15 and another not (e.g., x = 25*15 and y = 20) and in this case \(x^{16}-y^8\) won’t be divisible by 15. Not sufficient.

(2) \(y=x^2\)

Substitute y with \(x^2\) : \(x^{16}-y^8=x^{16}-(x^2)^8=x^{16}-x^{16}=0\) . 0 is divisible by 15. Sufficient

Notes for S1:

If integers a and b are both multiples of some integers k>1 then their sum and difference will also be a multiple of k
Example: a = 6 and b = 9, both divisible by 3: a + b = 15 and a – b = -3, again both divisible by 3.
If out of the integers a and b one is a multiple of some integers k>1 and another is not, then their sum and difference will NOT be a multiple of k
If integers a and b both are NOT multiples of some integer k>1, then their sum and difference may or may not be a multiple of k.
So according to above info that x is a multiple of 25, and y is a multiple of 20 tells us nothing whether \(x^{16}-y^8\) is divisible by 15.

Correct option: B

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