LCM

The least common multiple of the arguments.

✓ checked against that system · ! checked, with disagreements (hover a chip)WikipediaLeast common multiple↗MathWorldLeastCommonMultiple↗WikidataQ102761↗Encyclopedia of MathematicsLeast_common_multiple↗nLableast common multiple↗Britannicatopic/least-common-multiple↗FungrimLCM✓↗Fungrim entryWolfram LanguageLCM!↗SageMathlcm([$*,])!↗Rosetta CodeLeast common multiple↗

LCM(a, b, …)least common multiple of two or more integers.

Typed signatures

Type	Package	Applies
(any*) -> number	compute-engine	always
(any*) -> number	number-theoryreplaces compute-engine	always

Domain: Number theory

Details

• Also called the smallest common multiple: the smallest positive integer that is a multiple of every argument.
• Paired with GCD by $\gcd(a,b)\cdot\operatorname{lcm}(a,b)=ab$.
• compute-engine discards signs before computing, so $\operatorname{lcm}(-a,b)=\operatorname{lcm}(a,b)$.
• $\operatorname{lcm}(0,n)=0$: 0 absorbs, since 0 is a multiple of everything but nothing else divides back into it.
• Extends to rationals: $\operatorname{lcm}(p_1/q_1, \dots) = \operatorname{lcm}(p_1, \dots)/\gcd(q_1, \dots)$.

Examples

23 more kept as data, checked against the oracles

Basic #

live ↗

In$\operatorname{lcm}(4, 6)$Out$12$

LCM is variadic: it accepts any number of arguments

live ↗

In$\operatorname{lcm}(4, 6, 10)$Out$60$

live ↗

In$\operatorname{lcm}(2, 3, 5)$Out$30$

Scope #

Over the rationals: the lcm of the numerators over the gcd of the denominators — the smallest positive rational that is an integer multiple of each

live ↗

In$\operatorname{lcm}(\frac{1}{3}, \frac{2}{5}, \frac{3}{7})$Out$6$

$-1 + 3i = i(3 + i)$: associates share their multiples

live ↗

In$\operatorname{lcm}(3+i, -1+3i)$Out$3+i$

Negative integers

live ↗

In$\operatorname{lcm}(3, -5, 7)$Out$105$

Gaussian primes of norm 41 and 13: the LCM is the first-quadrant associate of their product

live ↗

In$\operatorname{lcm}(-4+5i, 2+3i)$Out$23+2i$

Gaussian rationals: $\operatorname{lcm}(5+6i,\,1+3i)/\gcd(10,3)$

live ↗

In$\operatorname{lcm}(\frac{1}{2}+\frac{3}{5}i, \frac{1}{3}+i)$Out$21+13i$

Big integers: the exact 118-digit LCM, past where a double loses precision

live ↗

In$\operatorname{lcm}(20!, 10^{100}+3)$Out$3\,475\,574\,297\,395\,200\,000\,000\,000\,000\,000\,000\,000\,000\,000\,000\,000\,000\,000\,000\,000\,000\,000\,000\,000\,000\,000\,000\,000\,000\,000\,000\,000\,001\,042\,672\,289\,218\,560\,000$

Listable: a single list argument threads element-wise against the rest

live ↗

In$\operatorname{lcm}(12, \bigl\lbrack3, 7, 40\bigr\rbrack)$Out$\bigl\lbrack12, 84, 120\bigr\rbrack$

A lazy Range, not a materialised List

live ↗

In$\operatorname{lcm}(1..10)$Out$2\,520$

Known value · Hand-computed: the smallest number divisible by each of 1 through 10 (same fact as the Applications row below, over List) -- an independent check that a lazy [[Range]] argument is walked element by element

live ↗

In$\mathrm{ReduceConditions}(\operatorname{lcm}(a, 3)\lt3a∧-10\lt a\lt10, a, ℤ)$Out$\mathrm{ReduceConditions}(\operatorname{lcm}(3, a)\lt3a∧-10\lt a\lt10, a, ℤ)$

Applications #

The smallest number divisible by each of 1 through 10

live ↗

In$\operatorname{lcm}(1, 2, 3, 4, 5, 6, 7, 8, 9, 10)$Out$2\,520$

Smallest common multiple of 10 and 15, e.g. for adding $\frac1{10}+\frac1{15}$

live ↗

In$\operatorname{lcm}(10, 15)$Out$30$

live ↗

In$\mathrm{FoldList}(\mathrm{LCM}, 1, 2..10)$Out$\bigl\lbrack1, 2, 6, 12, 60, 60, 420, 840, 2\,520, 2\,520\bigr\rbrack$

live ↗

In$\ln(\operatorname{lcm}(1..10))$Out$2\ln(3)+3\ln(2)+\ln(35)$

Properties #

$\gcd(a,b)\cdot\mathrm{lcm}(a,b)=ab$. See GCD

live ↗

In$\gcd(27, 81)\operatorname{lcm}(27, 81)=27×81$Out$\mathrm{True}$

When one number divides the other, the larger is the LCM

live ↗

In$\operatorname{lcm}(27, 81)$Out$81$

Accepts more than two arguments: $2\cdot3\cdot11\cdot5^2=1650$

live ↗

In$\operatorname{lcm}(6, 11, 25)=1\,650$Out$\mathrm{True}$

Argument order does not matter; coprime inputs multiply to $70$

live ↗

In$\operatorname{lcm}(5, 7, 2)=70$Out$\mathrm{True}$

live ↗

In$\ln(\frac{\operatorname{lcm}(1..5)}{\operatorname{lcm}(1..4)})$Out$\ln(5)$

Possible issues #

A single argument is returned unchanged

live ↗

In$\operatorname{lcm}(5)$Out$5$

0 absorbs: the LCM with 0 is 0

live ↗

In$\operatorname{lcm}(0, 5)$Out$0$

Signs are discarded before computing the LCM

live ↗

In$\operatorname{lcm}(-3, 7)$Out$21$

An infinite argument has no multiples to take: the call stays unevaluated, as in Wolfram

live ↗

In$\operatorname{lcm}(∞, 6)$Out$\operatorname{lcm}(6, ∞)$

About

Examples	48
Checked against	Wolfram Language 45 agree3 differSageMath 41 agree1 differ1 unevaluatedSymPy 20 agree13 differ2 unevaluatedmathlib4 9 agree2 errorOscar 5 agree5 errorJulia 9 agree26 errorRust 8 agree2 error
Crosswalk	WikipediaMathWorldWikidataEncyclopedia of MathematicsnLabBritannicaFungrimFungrim entry ×26Wolfram LanguageSageMath ×2SymPyOscarJuliaRustRosetta Code

Implementation

mappedwolframpackages/oracle/src/mappings.ts

compute-engine extends LCM to rationals (lcm of the numerators over the gcd of the denominators); Nemo/Oscar's ZZ conversion only accepts integers, so a rational operand here is a domain gap in the mapping, not a bug on either side.

mappedsympypackages/oracle/src/mappings.ts

compute-engine extends LCM to rationals (lcm of the numerators over the gcd of the denominators); Nemo/Oscar's ZZ conversion only accepts integers, so a rational operand here is a domain gap in the mapping, not a bug on either side.

mappedsagepackages/oracle/src/mappings.ts

compute-engine extends LCM to rationals (lcm of the numerators over the gcd of the denominators); Nemo/Oscar's ZZ conversion only accepts integers, so a rational operand here is a domain gap in the mapping, not a bug on either side.

mappedoscarpackages/oracle/src/mappings.ts

compute-engine extends LCM to rationals (lcm of the numerators over the gcd of the denominators); Nemo/Oscar's ZZ conversion only accepts integers, so a rational operand here is a domain gap in the mapping, not a bug on either side.

mappedjuliapackages/oracle/src/mappings.ts

compute-engine extends LCM to rationals (lcm of the numerators over the gcd of the denominators); Nemo/Oscar's ZZ conversion only accepts integers, so a rational operand here is a domain gap in the mapping, not a bug on either side.

mappedmathlib4packages/oracle/src/mappings.ts

compute-engine extends LCM to rationals (lcm of the numerators over the gcd of the denominators); Nemo/Oscar's ZZ conversion only accepts integers, so a rational operand here is a domain gap in the mapping, not a bug on either side.

mappedrustpackages/oracle/src/mappings.ts

compute-engine extends LCM to rationals (lcm of the numerators over the gcd of the denominators); Nemo/Oscar's ZZ conversion only accepts integers, so a rational operand here is a domain gap in the mapping, not a bug on either side.

mappedsagepackages/oracle/src/mappings.ts

See also: GCD