Elwyn D. answered • 03/21/16

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I am not certain how you want such a proof developed, but I can give you the general strategy.

Any number congruent to 2 (mod 4) is going to be congruent to either 2 (mod 8) or 6 (mod 8).

4 (mod 8) is congruent to 0 (mod 4).

2 (mod 4) is congruent to any number 2 + 4n for any integer n.

4 (mod 8) is congruent to any number 4 + 8m for any integer m.

4 (mod 8) is congruent to any number 4 + 8m for any integer m.

There is no pair of integers (n,m) such that

2 + 4n is congruent to 4 + 8m.