Mod¶

Status: Stable

documented, exercised by the test suite and/or worked examples, with no known limitations recorded.

Description¶

Mod[m, n]

gives the remainder of m on division by n, carrying the sign of the divisor n (a floored modulus, so Mod[-17, 5] is 3, not C's -2).

Mod[m, n, d]

gives the representative congruent to m modulo n in the half-open range d \<= r \< d + n.

Notes

Reduction is exact on Integer / BigInt / Rational inputs (no overflow), and works for Real / MPFR arguments at the input precision. Mod is Listable; non-numeric arguments are left unevaluated.

Examples (4)¶

Every input below was run against the current Mathilda build and its output recorded.

Applications (4)¶

In[1]:= Mod[-17, 5]
Out[1]= 3

In[2]:= Mod[17, -5]
Out[2]= -3

In[3]:= Mod[2^100, 7]
Out[3]= 2

In[4]:= Mod[10, 3, 1]
Out[4]= 1

Implementation notes¶

builtin_mod handles the two- and three-argument forms. For Integer and BigInt operands it reduces with GMP's mpz_fdiv_r — a floored remainder, so the result carries the sign of the divisor n; a rational path computes m - n*Floor[m/n] exactly in mpq; MPFR operands compute m - n*floor(m/n) at the larger of the two input precisions; and a machine-double fallback covers mixed Integer/Real arguments. The three-argument Mod[m, n, d] reduces m - d and adds d back, landing the representative in the half-open range [d, d+n). Mod is registered PROTECTED | NUMERICFUNCTION | LISTABLE and runs element-wise on a packed machine buffer. Non-numeric arguments return NULL (left symbolic).

Attributes: Listable, NumericFunction, Protected.

References¶

See also: Quotient, QuotientRemainder, Union, Tally, DeleteDuplicates

Knuth, "The Art of Computer Programming, Vol. 2: Seminumerical Algorithms", on the division algorithm and modular reduction.
von zur Gathen & Gerhard, "Modern Computer Algebra", on residues and modular arithmetic.
Source: src/core.c
Specification: docs/spec/builtins/arithmetic.md
Tests: tests/test_association.c
Tests: tests/test_bigint.c
Tests: tests/test_compile.c
Tests: tests/test_compile_coverage.c

Notes & additional examples¶

Notes¶

Mod[m, n] returns a residue that takes the sign of the divisor n, so Mod[-17, 5] = 3 (non-negative) while Mod[17, -5] = -3 (non-positive). This is the mathematician's floored modulus, not C's truncated %. Reduction is exact on bigints, so Mod[2^100, 7] = 2 without overflow. The three-argument form Mod[m, n, d] returns the representative in the offset range [d, d+n), e.g. Mod[10, 3, 1] = 1.