[Geometry]
Problem: Rectangle $A R M L$ has length 125 and width 8. The rectangle is divided into 1000 squares of area 1 by drawing in gridlines parallel to the sides of $A R M L$. Diagonal $\overline{A M}$ passes through the interior of exactly $n$ of the 1000 unit squares. Compute $n$.

Solution: Notice that 125 and 8 are relatively prime. Examining rectangles of size $a \times b$ where $a$ and $b$ are small and relatively prime suggests an answer of $a+b-1$. To see that this is the case, note that other than the endpoints, the diagonal does not pass through any vertex of any unit square. After the first square, it must enter each subsequent square via a vertical or horizontal side. By continuity, the total number of these sides is the sum of the $a-1$ interior vertical lines and $b-1$ interior horizontal lines. The diagonal passes through $(a-1)+(b-1)=a+b-2$ additional squares, so the total is $a+b-1$. Because 125 and 8 are relatively prime, it follows that $N=125+8-1=\mathbf{1 3 2}$.

Remark: As an exercise, the reader is encouraged to show that the answer for general $a$ and $b$ is $a+b-\operatorname{gcd}(a, b)$.

Final answer: 132