32 Given that $n$ is a ten-digit number in the form $\overline{2007 x 2008 y}$ where $x$ and $y$ can be any of the digits $0,1,2, \ldots, 9$. How many such numbers $n$ are there that are divisible by 33 ?
32. Answer: 3
$33=3 \times 11$. So 3 must divide $2+7+x+2+8+y \Rightarrow 19+x+y=3 k_{1} .11$ must divide $(7+2+y)-(2+x+8) \Rightarrow y-x-1=11 k_{2} \Rightarrow-10 \leq 11 k_{2} \leq 8 \Rightarrow k_{2}=0$. So $y=x+1 \Rightarrow 20+2 x=3 k_{1}$. When $k_{1}=8, x=2$ and $y=3$. When $k_{1}=10, x=5$ and $y=6$. When $k_{1}=12, x=8$ and $y=9$.