What a lot of work japhy! Here's a short version.

Let N be an integer that is divisible by 7

Let "xy" be a number .
Eg, for 203,  x=20, y=3
    for 1984, x=198 y=4

If this number is divisible by seven, then:

10x + y = N

=> 7x + 3x + y = N

=> 3x + y = N

=> 9x + 3y = N

So

10x + y = N
9x + 3y = N

=> x - 2y = N
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ie, if "xy" is divisible by 7, then x-2y is also divisible by 7.

And, if you can be bothered, it can easily be shown that the reverse is true also, so we have:

"xy" % 7 = 0  <=>  (x - 2y) % 7 = 0
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cLive ;-)