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

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. 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 (as we cannot factor out 15 from \(x^{16}-y^8\)). 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 (zero is divisible by every integer except zero itself). Sufficient.

Notes for statement (1): If integers \(a\) and \(b\) are both multiples of some integer \(k \gt 1\) (divisible by \(k\)), then their sum and difference will also be a multiple of \(k\) (divisible by \(k\)):

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 integers \(a\) and \(b\) one is a multiple of some integer \(k \gt 1\) and another is not, then their sum and difference will NOT be a multiple of \(k\) (divisible by \(k\)):

Example: \(a=6\), divisible by 3 and \(b=5\), not divisible by 3: \(a+b=11\) and \(a-b=1\), neither is divisible by 3.

If integers \(a\) and \(b\) both are NOT multiples of some integer \(k \gt 1\) (divisible by \(k\)), then their sum and difference may or may not be a multiple of \(k\) (divisible by \(k\)):

Example: \(a=5\) and \(b=4\), neither is divisible by 3: \(a+b=9\), is divisible by 3 and \(a-b=1\), is not divisible by 3

OR: \(a=6\) and \(b=3\), neither is divisible by 5: \(a+b=9\) and \(a-b=3\), neither is divisible by 5

OR: \(a=2\) and \(b=2\), neither is divisible by 4: \(a+b=4\) and \(a-b=0\), both are divisible by 4.

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.

Answer: B

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