Least Common Multiple Calculator

Integer arithmetic

Least Common Multiple Calculator

Enter two or more integers. For nonzero inputs, the smallest positive value divisible by every entered number appears first, followed by a transparent factorization and exact-division check. Lists containing zero follow the stated zero convention.

Input

Integers to compare

Separate values with commas, spaces, semicolons, or new lines.

Result

Least common multiple

The answer stays prominent; the proof follows below.

Least common multiple of 15, 20

Calculated from 2 entered integers using exact whole-number arithmetic.

Method

See how the answer is built

Prime factorization

Keep every prime at its highest required power

153 × 5
202² × 5

Highest prime powers3 × 5 × 2² = 60

Greatest common divisor reduction

Divide before multiplying, then carry the result into the next pair.

Pair 1: least common multiple of 15 and 20 = 60Greatest common divisor = 5(15 ÷ 5) × 20 = 60

Exact division check

Every quotient is an integer, so the answer is a common multiple of every entered value.

60 ÷ 15 = 4Whole number
60 ÷ 20 = 3Whole number

Integer arithmetic guide

How to Find the Least Common Multiple

The least common multiple of two or more positive integers is the smallest positive integer divisible by every one of them. A clear solution should give the answer and show why no required prime power has been omitted.

Side-by-side guide · Integer arithmetic guide

Three reliable ways to find a common multiple

Definition: Smallest shared positive multiple
Prime rule: Keep each highest required exponent
Worked answer: 330, 75, 450, 225 → 4,950

List multiples for small inputs
Write successive positive multiples of each number and stop at the first value shared by every list. This is easy to inspect for small pairs such as 18 and 24, but the lists become long for larger or numerous inputs.
Compare prime factorizations
Factor every positive magnitude, then keep each prime at the highest exponent found in any one factorization. Multiplying those selected prime powers creates the smallest number containing every divisibility requirement.
Reduce with the greatest common divisor
For two nonzero integers a and b, divide one magnitude by their greatest common divisor before multiplying by the other. Apply the same identity repeatedly when the list contains more than two integers.

How to Find the Least Common Multiple: The least common multiple of two or more positive integers is the smallest positive integer divisible by every one of them. A clear solution should give the answer and show why no required prime power has been omitted.

What is the least common multiple?

Multiples move upward from a number: the positive multiples of 18 begin 18, 36, 54, 72, and 90, while those of 24 begin 24, 48, 72, and 96. The first shared entry is 72, so 72 is the least common multiple of 18 and 24. Later shared entries such as 144 are common multiples, but they are not the least one.

This is different from a greatest-common-factor question. A common multiple must be divisible by every input, while a common factor must divide every input. Use the greatest common factor calculator when the goal is the largest exact group or cut size rather than the first shared multiple.

Three reliable ways to find a common multiple

List multiples for small inputs

Write successive positive multiples of each number and stop at the first value shared by every list. This is easy to inspect for small pairs such as 18 and 24, but the lists become long for larger or numerous inputs.

Compare prime factorizations

Factor every positive magnitude, then keep each prime at the highest exponent found in any one factorization. Multiplying those selected prime powers creates the smallest number containing every divisibility requirement.

Reduce with the greatest common divisor

For two nonzero integers a and b, divide one magnitude by their greatest common divisor before multiplying by the other. Apply the same identity repeatedly when the list contains more than two integers.

Worked example: 330, 75, 450, and 225

Factor every integer
Write 330 = 2×3×5×11, 75 = 3×5², 450 = 2×3²×5², and 225 = 3²×5². Keeping repeated factors as exponents makes the comparison easier to audit; use the factor calculator when you want a separate prime-factor trace for one of the inputs.
Keep the highest power of every prime
The inputs require 2¹, 3², 5², and 11¹. Choosing a lower power would make the result fail to divide at least one input; choosing an extra factor would produce a common multiple that is not the least one.
Multiply and verify
Calculate 2×3²×5²×11 = 4,950. The checks 4,950÷330 = 15, 4,950÷75 = 66, 4,950÷450 = 11, and 4,950÷225 = 22 confirm that every quotient is an integer.

Greatest-common-divisor reduction reaches the same answer

For two nonzero values, use |a÷greatest common divisor(a,b)×b|. Dividing first limits intermediate growth without changing the exact integer result. For 21 and 14, the greatest common divisor is 7, so (21÷7)×14 = 42.

For a longer list, carry the result forward. The greatest common divisor of 42 and 38 is 2, so (42÷2)×38 = 798. Therefore the least common multiple of 21, 14, and 38 is 798. This pairwise reduction is the governed method used by the calculator, while the prime-factor display provides a second way to inspect modest inputs.

Use the result with the right assumptions

Recurring schedules: if inspections repeat every 18 and 24 days from the same starting point, their next shared cycle is after 72 days. Different starting offsets require a separate congruence calculation.
Fraction denominators: a least common multiple can supply a least common denominator, but each numerator must be scaled by the same factor as its denominator. The fraction calculator keeps those equivalent fractions together.
Units and meaning: convert all intervals to the same unit before calculation. A mathematically correct multiple is not automatically a workable calendar date, package size, or operating schedule.
Signs and zero: the calculator uses integer magnitudes, so negative signs do not change divisibility. It explicitly returns zero when any input is zero, although zero is not a usable positive repeating interval and has no prime factorization.
Input limits: enter 2–100 safe integers. Decimal values are not integer multiple problems, and an exact nonzero result beyond the supported safe-integer range is rejected rather than rounded.

Important limitations

Inputs must be safe integers. The calculator defines any list containing zero to have a least common multiple of zero, ignores signs for divisibility, and rejects a nonzero exact result outside the safe integer range instead of rounding it.

Review the stated scope, calculation method, assumptions, supported inputs, and effective dates, and obtain qualified advice before using a high-stakes result.

This formula passed its registered engineering tests and is published under an explicit site-owner policy authorization. No independent expert, QA, editorial, legal, or third-party review is claimed. Verify consequential results with a qualified professional.

Integer arithmetic guide

How to Find the Least Common Multiple

Side-by-side guide · Integer arithmetic guide

Three reliable ways to find a common multiple

Definition: Smallest shared positive multiple
Prime rule: Keep each highest required exponent
Worked answer: 330, 75, 450, 225 → 4,950

List multiples for small inputs
Write successive positive multiples of each number and stop at the first value shared by every list. This is easy to inspect for small pairs such as 18 and 24, but the lists become long for larger or numerous inputs.
Compare prime factorizations
Factor every positive magnitude, then keep each prime at the highest exponent found in any one factorization. Multiplying those selected prime powers creates the smallest number containing every divisibility requirement.
Reduce with the greatest common divisor
For two nonzero integers a and b, divide one magnitude by their greatest common divisor before multiplying by the other. Apply the same identity repeatedly when the list contains more than two integers.

What is the least common multiple?

Three reliable ways to find a common multiple

List multiples for small inputs

Compare prime factorizations

Reduce with the greatest common divisor

Worked example: 330, 75, 450, and 225

Factor every integer
Write 330 = 2×3×5×11, 75 = 3×5², 450 = 2×3²×5², and 225 = 3²×5². Keeping repeated factors as exponents makes the comparison easier to audit; use the factor calculator when you want a separate prime-factor trace for one of the inputs.
Keep the highest power of every prime
The inputs require 2¹, 3², 5², and 11¹. Choosing a lower power would make the result fail to divide at least one input; choosing an extra factor would produce a common multiple that is not the least one.
Multiply and verify
Calculate 2×3²×5²×11 = 4,950. The checks 4,950÷330 = 15, 4,950÷75 = 66, 4,950÷450 = 11, and 4,950÷225 = 22 confirm that every quotient is an integer.

Greatest-common-divisor reduction reaches the same answer

Use the result with the right assumptions

Recurring schedules: if inspections repeat every 18 and 24 days from the same starting point, their next shared cycle is after 72 days. Different starting offsets require a separate congruence calculation.
Fraction denominators: a least common multiple can supply a least common denominator, but each numerator must be scaled by the same factor as its denominator. The fraction calculator keeps those equivalent fractions together.
Units and meaning: convert all intervals to the same unit before calculation. A mathematically correct multiple is not automatically a workable calendar date, package size, or operating schedule.
Signs and zero: the calculator uses integer magnitudes, so negative signs do not change divisibility. It explicitly returns zero when any input is zero, although zero is not a usable positive repeating interval and has no prime factorization.
Input limits: enter 2–100 safe integers. Decimal values are not integer multiple problems, and an exact nonzero result beyond the supported safe-integer range is rejected rather than rounded.