Birds of a Feather

A problem from the February 2006 issue of Crux Mathematicorum:

Prove that if 10a + b is a multiple of 7 then a – 2b must be a multiple of 7 as well.

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This solution is by Eric Zhang. If 10a + b is a multiple of 7, then 10a + b = 7n, where n is an integer. Rearrange that as b = 7n – 10a and substitute it into a – 2b and you get a – 2b = a – 2(7n – 10a) = 21a – 14n, which is a multiple of 7.