Modular arithmetic with negative numbers

Question

Is this correct

-347 mod 6 = -5

Or is this correct

-347 mod 6 = +1

Answer 1

There's a slight subtlety here: we usually let $a$ mod $b$ be the least nonnegative $k$ such that $a=k+nb$ for some integer $n$; thus, we would not say $-347$ mod $6$ = $-5$. Here, we're using mod $6$ as a function from integers to integers.

However, it is true that $-347, -5,$ and $1$ are all equal mod $6$. This is a slightly different use of mod $6$: we're viewing mod $6$ as an equivalence relation on integers.

This should help explain why e.g. computer algebra systems say $-347$ mod $6=1$, and not $-5$. In particular, both are true most of the time, but if we're thinking of mod as a function which spits out the "canonical" answer, then the second is right and the first isn't.