Better Late Than Never... Problem Solving on Diagonals

It seemed simple when I first received this problem in February, but I found out from a couple of other students that it's not as simple as it seemed. Might as well get this out before the end of the course.

Problem: Draw a diagonal line in a rectangle that consists of m rows and n columns and determine how many grid squares are passed through by the diagonal line.

Process: Using graph paper in class, I drew some examples down on paper and found some patterns.
This is what I found comparing m and n (before I ran out of space).

    n 
m       1  2  3  4  5  6  7  8

    1    1  2  3  4  5  6  7  8              RED: Whenever n = m
    2    2  2  4  4  6  6  8  8              GREEN: Whenever n = m are odd
    3    3  4  3  6  7  6  9 10             YELLOW: Whenever n and m are odd
    4    4  4  6  4  8  8 10 8
    5    5  6  7  8  5 101112
    6    6  6  6  8 10 6 1212
    7    7  8  9 101112 7 14
    8    8  8 10 8 121214 8

I wasn't so sure about the pattern that I found, because it seemed to look like it was all over the place.
I initially thought that there was a pattern whenever n or m are odd or whenever n or m are even, but then I had to look at whenever n = m, where the pattern would break. There also seems to be a pattern when looking diagonally across this chart.

Overall, I'm not entirely sure of what to make of this, even though "answer" were already given. I felt that if possible, I'd want to discuss this with Danny about this, but it's already the end of the course, so oh well.