Mod

The remainder of a on division by b.

✓ checked against that system · ! checked, with disagreements (hover a chip)WikipediaModulo↗MathWorldMod↗WikidataQ1799665↗DLMFfront/introduction.common.t2.r10↗FungrimMod✓↗Fungrim entry540931↗75231e↗959a25✓↗Wolfram LanguageMod!↗

Mod(a, b)remainder of $a$ on division by $b$, with the sign of $b$.

SageMathenumeratio_mod(a, b)Oscarmod(a, b)Juliamod(a, b)Rustmod_floor(a, b)

Mod(a, b, d)the $x \equiv a \pmod b$ with $d \le x < d + b$.

Mod(z, m)for Gaussian integers, $z - m\,\mathrm{Quotient}(z, m)$: the remainder in the box around 0

SageMathenumeratio_mod(z, m)Oscarmod(z, m)Juliamod(z, m)Rustmod_floor(z, m)

Mod(a, b, d?)The remainder of a on division by b.

Typed signatures

Type	Package	Applies
(real, real) -> real	analyticreplaces compute-engine	always
(real, real) -> real	compute-engine	always
(real, real) -> real	residuesreplaces analytic	always
(number, number, number?) -> number	number-theoryreplaces residues	always

Domain: Modular arithmetic

Details

• Equivalent to $a-b\,\mathrm{Quotient}(a,b)$, i.e. $a-b\lfloor a/b\rfloor$.
• When $b>0$ the result lies in $[0,b)$; the sign of the result always matches the sign of $b$.
• Periodic: $a\bmod n=(a+kn)\bmod n$ for any integer $k$.
• compute-engine returns NaN for a zero modulus rather than leaving the call unevaluated.
• A third argument $d$ offsets the range to $[d, d+b)$, as Wolfram's Mod[a, b, d] does.
• Sage's Mod(a, m) builds a residue class, which is IntegerMod here (IntegerMod(a, m) keeps its modulus); Mod(a, m) is the remainder, a plain number, as in Wolfram. Sage's a % m is this head.

Examples

30 more kept as data, checked against the oracles

Basic #

live ↗

In$17\bmod5$Out$2$

live ↗

In$100\bmod7$Out$2$

The result takes the sign of the modulus

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In$(-7)\bmod3$Out$2$

Threads element-wise over a list

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In$\bigl\lbrack1, 2, 3, 4, 5\bigr\rbrack\bmod3$Out$\bigl\lbrack1, 2, 0, 1, 2\bigr\rbrack$

The remainder of 5 divided by 3

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In$5\bmod3$Out$2$

Mod binds tighter than addition, so this is $1 + (4 \bmod 3)$

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In$1+4\bmod3$Out$2$

Sage's Mod(17, 5) is a residue class, IntegerMod; Mod(17, 5) here is the remainder 2, a plain integer

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In$({17}\;\mathrm{mod}\;5)$Out$(2\;\mathrm{mod}\;5)$

Scope #

A third argument offsets the range: the result lies in $[d, d + n)$, here $[1, 6)$

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In$\mathrm{Mod}(17, 5, 1)$Out$2$

With offset 1 the residues run $1, 2, 3$ rather than $0, 1, 2$

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In$\mathrm{Mod}(5, 3, 1)$Out$2$

Rationals are reduced exactly

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In$\frac{5}{2}\bmod2$Out$\frac{1}{2}$

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In$3.14\bmod2$Out$1.14$

An exact irrational reduces exactly to $2\sqrt7 - 3$

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In$\sqrt{28}\bmod3$Out$2\sqrt{7}-3$

A symbolic constant reduces exactly to $\pi - 2$

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In$π\bmod2$Out$π-2$

a 10 001-digit dividend

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In$10^{10\,000}\bmod10\,007$Out$6\,333$

Threads over a list of moduli

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In$10\bmod\bigl\lbrack3, 4, 7\bigr\rbrack$Out$\bigl\lbrack1, 2, 3\bigr\rbrack$

a Gaussian integer: $(5 + 3i) - 2(2 + 2i)$

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In$(5+3i)\bmod2$Out$1-i$

$1 + i$ divides $5 + 3i = (1 + i)(4 - i)$

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In$(5+3i)\bmod(1+i)$Out$0$

Gaussian integers: the quotient rounds, so each part lands in $(-m/2, m/2]$

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In$(7+5i)\bmod3$Out$1-i$

a Gaussian modulus: $7 + 5i = (2 + i)(4 + i) - i$

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In$(7+5i)\bmod(2+i)$Out$-i$

exact in both parts past $2^{53}$

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In$(100\,000\,000\,000\,000\,000\,001+9\,007\,199\,254\,740\,993i)\bmod(2+i)$Out$0$

Again $1 + (16 \bmod 5)$ from precedence

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In$1+16\bmod5$Out$2$

The generating function of $n \bmod 4$ is rational

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In$\mathrm{GeneratingFunction}(n\bmod4, n, x)$Out$\frac{3x^3+2x^2+x}{1-x^4}$

Applications #

$282475249 = 7^{10}$, so Fermat's little theorem gives 1

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In$282\,475\,249\bmod11$Out$1$

Euler's theorem: $7^{\varphi(12)} \equiv 1 \pmod{12}$

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In$7^{\mathrm{Totient}(12)}\bmod12$Out$1$

Cycling an index with Mod repeats the list

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In$\bigl\lbrack a, b, c\bigr\rbrack[1+((1..10)-1)\bmod3]$Out$\bigl\lbrack a, b, c, a, b, c, a, b, c, a\bigr\rbrack$

Properties #

Periodicity: $a \bmod n = (a+kn) \bmod n$ for any integer $k$

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In$17\bmod5=(17+3×5)\bmod5$Out$\mathrm{True}$

A negative modulus gives a result in $(n, 0]$

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In$(-5)\bmod(-3)$Out$-2$

the remainder of Quotient

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In$(7+5i)\bmod(2+i)=(7+5i)-(2+i)\mathrm{Quotient}(7+5i, 2+i)$Out$\mathrm{True}$

Congruent numbers share a remainder

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In$5\bmod3=2\bmod3$Out$\mathrm{True}$

$2 \equiv 8 \pmod 3$

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In$2\bmod3=8\bmod3$Out$\mathrm{True}$

Anything is 0 modulo 1

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In$0\bmod1$Out$0$

Known value · Wolfram MathematicalFunctionData, Mod

Odd numbers leave remainder 1 modulo 2

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In$1\bmod2$Out$1$

Known value · Wolfram MathematicalFunctionData, Mod

Remainder 1 at the start of the cycle

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In$1\bmod3$Out$1$

Known value · Wolfram MathematicalFunctionData, Mod

The remainders modulo 3 cycle through 0, 1, 2

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In$2\bmod3$Out$2$

Known value · Wolfram MathematicalFunctionData, Mod

A multiple of the modulus leaves 0

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In$3\bmod3$Out$0$

Known value · Wolfram MathematicalFunctionData, Mod

The cycle repeats after 3

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In$4\bmod3$Out$1$

Known value · Wolfram MathematicalFunctionData, Mod

$8 \equiv 5 \pmod 3$, both leave 2

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In$8\bmod3=5\bmod3$Out$\mathrm{True}$

An even number has remainder 0 modulo 2

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In$8\bmod2$Out$0$

A larger remainder, $17 = 2\cdot 6 + 5$

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In$17\bmod6$Out$5$

The remainder is smaller than the modulus 8

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In$12\bmod8$Out$4$

Known value · Wolfram MathematicalFunctionData, Mod

Negative operands take the sign of the divisor

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In$(-3)\bmod(-2)$Out$-1$

Known value · Wolfram MathematicalFunctionData, Mod

Rational operands work too: $-\frac{27}{10} + 2\cdot\frac{23}{5} = \frac{19}{10}$

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In$-\frac{27}{10}\bmod\frac{23}{5}$Out$\frac{19}{10}$

Known value · Wolfram MathematicalFunctionData, Mod

Zero is zero modulo anything

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In$0\bmod2$Out$0$

Known value · Wolfram MathematicalFunctionData, Mod

Every integer is 0 modulo 1

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In$1\bmod1$Out$0$

Known value · Wolfram MathematicalFunctionData, Mod

$15 = 2\cdot 7 + 1$

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In$15\bmod7$Out$1$

Works for reals: $\pi \bmod e = \pi - e$ since $\pi < 2e$

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In$π\bmod e$Out$π-e$

Known value · Wolfram MathematicalFunctionData, Mod

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In$2^{\mathrm{Totient}(3)}\bmod3$Out$1$

Known value · Wolfram MathematicalFunctionData, Mod, Euler's totient theorem

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In$(1+(\mathrm{NthPrime}(2)-1)!)\bmod\mathrm{NthPrime}(2)$Out$0$

Known value · Wolfram MathematicalFunctionData, Mod, Wilson's theorem

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In$(2-\mathrm{JacobiSymbol}(2, 3))\bmod3$Out$0$

Known value · Wolfram MathematicalFunctionData, Mod, Euler's criterion

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In$(1+(\mathrm{NthPrime}(3)-1)!)\bmod\mathrm{NthPrime}(3)$Out$0$

Known value · Wolfram MathematicalFunctionData, Mod, Wilson's theorem

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In$2π\bmod e$Out$2π-2e$

Known value · Wolfram MathematicalFunctionData, Mod

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In$-π\bmod2$Out$4-π$

Known value · Wolfram MathematicalFunctionData, Mod

Possible issues #

Division by a 0 modulus yields Indeterminate rather than an error

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In$5\bmod0$Out$\operatorname{Indeterminate}$

rounding ties to even means $1$ and $-1$ both appear as remainders mod 2

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In$\bigl\lbrack(1+2i)\bmod2, (3+2i)\bmod2\bigr\rbrack$Out$\bigl\lbrack1, -1\bigr\rbrack$

An infinite dividend has no remainder

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In$∞\bmod3$Out$\operatorname{NaN}$

An infinite modulus gives NaN: compute-engine takes no remainder against an infinity

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In$5\bmod∞$Out$\operatorname{NaN}$

ComplexInfinity as the dividend leaves the call unevaluated

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In$\tilde∞\bmod3$Out$\tilde∞\bmod3$

About

Examples	87
Checked against	Wolfram Language 83 agree3 differSageMath 21 agree3 errorSymPy 48 agree17 differ12 unevaluated5 errormathlib4 4 agree1 unevaluated1 errorOscar 9 agree10 errorJulia 10 agree8 errorRust 7 agree9 error
Crosswalk	WikipediaMathWorldWikidataDLMFFungrimFungrim entry ×3Wolfram LanguageSageMathOscarJuliaRust

Implementation

mappedwolframpackages/oracle/src/mappings.ts

mappedsympypackages/oracle/src/mappings.ts

mappedsagepackages/oracle/src/mappings.ts

mappedoscarpackages/oracle/src/mappings.ts

mappedjuliapackages/oracle/src/mappings.ts

mappedmathlib4packages/oracle/src/mappings.ts

mappedrustpackages/oracle/src/mappings.ts

See also: PowerMod, IntegerMod