For any n x m rectangle such that GCD(n,m) = 1, find a rule determining the number of unit squares (1 x 1) that a diagonal passes through, Math Help

Apr 26th, 2016
Steve1995
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Question description

For any n x m rectangle such that GCD(n,m) = 1, find a rule determining the number of unit squares (1 x 1) that a diagonal passes through. For example the diagonal passes through 8 and 6 unit squares, respectively.

Tutor Answer

(Top Tutor) Daniel C.
(997)
School: Rice University
PREMIUM TUTOR

HeyHere is the work.

SOLUTION
The best way to approach this problem seems to start with a few small rectangles 2×3, 2×5, 3×5,
etc. Tabulate the results and next try to generalize. There are going to be exceptional cases, like
2×4, 3×6, 4×6. These would need a special treatment and further generalization.

How can we count the crossed squares in that (simple) case? The upper left square is always
crossed. This is 1. Next, on its way to the next square the diagonal crosses one of the (internal)
grid lines. In fact, this happens every time we move f...

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