If $x$ and $y$ are integers with $(y-1)^{x+y}=4^{3}$, then the number of possible values for $x$ is
(A) 8
(B) 3
(C) 4
(D) 5
(E) 6
The number $4^{3}$ equals 64 .

To express 64 as $a^{b}$ where $a$ and $b$ are integers, we can use $64^{1}, 8^{2}, 4^{3}, 2^{6},(-2)^{6}$, and $(-8)^{2}$. We make a table to evaluate $x$ and $y$ :

| $y-1$ | $x+y$ | $y$ | $x$ |
| :---: | :---: | :---: | :---: |
| 64 | 1 | 65 | -64 |
| 8 | 2 | 9 | -7 |
| 4 | 3 | 5 | -2 |
| 2 | 6 | 3 | 3 |
| -2 | 6 | -1 | 7 |
| -8 | 2 | -7 | 9 |

Therefore, there are 6 possible values for $x$.

ANSWER: (E)