If \(a, b\), and \(c\) are integers, what is the value of \(a\)?

We're given no information in the question stem except that these variables are integers. So, we have to dive into the statements:

(1) \(2^a+2^b=33\)
Think of combinations of powers of 2 that would add to 33. Since 33 is odd, it must be (odd + even) or (even + odd). The only power of 2 that's odd is \(2^0=1\) .
\(2^0+2^5=1+32=33\)
We know that one of these values must be 0 and the other 5, but we don't know which is which. Insufficient.

(2) \(a\cdot c = 5\)
If both of these are integers, it must be 1*5 or 5*1. Since we don't know which is which, though, this is insufficient.

(1) and (2) together:
(1) tells us that \(a=0\) or \(a=5\), and (2) tells us that \(a=1\) or \(a=5\). Using the statements together, it must be the case that \(a=5\). Sufficient.

The answer is C.