7. $[7]$ A student at Harvard named Kevin
Was counting his stones by 11
He messed up $n$ times
And instead counted $9 \mathrm{~s}$
And wound up at 2007.
How many values of $n$ could make this limerick true?
Answer: 21. The mathematical content is that $9 n+11 k=2007$, for some nonnegative integers $n$ and $k$. As $2007=9 \cdot 223, k$ must be divisible by 9 . Using modulo 11 , we see that $n$ is 3 more than a multiple of 11. Thus, the possibilities are $n=223,212,201, \ldots, 3$, which are 21 in number.