If x and y are positive integers, is x^16 - y^8 + 345y^2 divisible by 15?

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

First of all as \(345y^2\) is divisible by 15 (this term won't affect the remainder), we can drop it.

The question becomes: is \(x^{16}-y^8\) divisible by 15?

(1) \(x\) is a multiple of 25, and \(y\) is a multiple of 20 --> they both 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 (eg x=25*15 and y=20) and in this case \(x^{16}-y^8\) won't be divisible by 15 (as we can not factor out 15 from \(x^{16}-y^8\)). Not sufficient.

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

Answer: B.



If x and y both are not multiples of 3, then their sum or difference may or may not be multiple of 3:

x=2 and y=1 --> x+y=3 --> sum is multiple of 3;
x=2 and y=2 --> x-y=0 --> difference is multiple of 3;
x=2 and y=3 --> x+y=5 --> sum is not multiple of 3;
x=2 and y=0 --> x-y=2 --> difference is not multiple of 3.

BUT, if x is multiple of 3 and y is not (or vise-versa), then their sum or difference won't be multiple of 3.

Hope it helps.
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From expression 345y^2 is certainly divisible 15 as 345/15 gives remainder 0.
we need to find if x^16 - y^8 is divisible by 15 or not.

From Statement 1:
If x=25 and y=20, then 25^16 - 20^8 = 5^8(5^24 - 4^8)/15 gives the remainder 10 ---> Not divisible
If x=75 and y=60, then 75^16 - 60^8 = 15^8(5^16 * 15^8 - 4^8)/15 gives the remainder 0 ---> divisible
Inconsistent result, so ST1 alone not sufficient.

From Statement 2:
y = x^2
(x^16 - x^16 + 345y^2) / 15 gives remainder 0 ---> Divisible.
so ST2 alone is sufficient.

Hence correct answer is B