Using the standard mathematical convention where the result is the least nonnegative residue, what is -17 mod 5?

-17 mod 5 = 3.

Under the least-nonnegative-residue convention, a mod n is the unique remainder r satisfying 0 ≤ r < n in the division-algorithm decomposition a = q·n + r, where q is an integer (the quotient).

For a = -17, n = 5:

• Choose the quotient by rounding -17/5 = -3.4 down to the nearest integer (floor): q = -4.
• Then r = a − q·n = -17 − (-4)(5) = -17 + 20 = 3.

Check: -17 = 5·(-4) + 3, and 3 lies in {0, 1, 2, 3, 4}, the complete set of least nonnegative residues mod 5.

Why not -2? The value -2 is congruent to -17 mod 5 (they differ by 15 = 3·5), but -2 is negative, so it is not the least nonnegative residue. -2 is what languages using truncated division (e.g., C, Java, Go for %) return for -17 % 5. Languages using floored division (Python's %, or mod in Ruby) return 3, matching the mathematical convention.

Quick verification in Python:

>>> -17 % 5
3

Sources:

• https://en.wikipedia.org/wiki/Modular_arithmetic
• https://www.southampton.ac.uk/~wright/1001/modular-arithmetic.html
• https://math.libretexts.org/Bookshelves/Combinatorics_and_Discrete_Mathematics/Elementary_Number_Theory_(Clark)/01:_Chapters/1.20:_Zm_and_Complete_Residue_Systems
• https://crypto.stanford.edu/pbc/notes/numbertheory/arith.html