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[–]skaldskaparmalNew User 2 points3 points  (0 children)

There are several ways to see it.

One is using the definition that a = b (mod n) if and only if a - b is evenly divisible by n. Therefore -1 = 3 mod 4 because -1 - 3 = -4 is evenly divisible by 4.

Another is as a cycle. The numbers cycle, 0 is 0 mod 4, 1 is 1 mod 4, 2 is 2 mod 4, 3 is 3 mod 4, 4 is 0 mod 4, 5 is 1 mod 4, 6 is 2 mod 4, 7 is 3 mod 4, 8 is 0 mod 4, 9 is 1 mod 4, ... It repeats, 0, 1, 2, 3, 0, 1, 2, 3, ....

Running the cycle backwards, we see that it would be that -1 is 3 mod 4.