How this instrument works
The least common multiple (LCM) of two whole numbers is the smallest positive whole number that both original numbers divide into evenly. While every pair of numbers shares infinitely many common multiples (their product is always one of them), the LCM is specifically the smallest, and it's found here via a clean identity linking it to the greatest common divisor: lcm(a, b) = (a × b) ⁄ gcd(a, b), avoiding the need to list out multiples one at a time until a match appears.
The LCM is exactly the smallest common denominator needed to add or subtract two fractions with different denominators. Using any OTHER common denominator (such as the plain product of the two, which is always also a valid common denominator) still works arithmetically, but leaves the result needing further simplification afterward — using the LCM specifically produces a sum that's already in its simplest form.
When two numbers happen to be coprime (share no common factor beyond 1), their GCD is 1, and the identity linking GCD and LCM shows their LCM must simply be their plain product — the smallest common multiple two unrelated numbers can share is nothing smaller than multiplying them together directly.
- Enter the first whole number into the First number field.
- Enter the second whole number into the Second number field.
- Read Least common multiple: the smallest number both divide into evenly.
Worked example — the LCM of 48 and 18
The least common multiple of 48 and 18 is 144 — check that 144 ⁄ 48 = 3 and 144 ⁄ 18 = 8, both exact, and no smaller number is divisible by both. This is exactly the common denominator that would be needed to add the fractions 1⁄48 and 1⁄18 without any further simplifying afterward.
Compare 17 and 5, two numbers sharing no common factor beyond 1: their LCM is 85, simply their plain product, since coprime numbers always have an LCM equal to a × b — nothing smaller can be a multiple of both when they share no other structure.
Questions
What is the least common multiple?
The LCM of two whole numbers is the smallest positive whole number that both numbers divide into evenly. Any common multiple works for some purposes, but the LCM specifically is the smallest one, avoiding unnecessary extra size.
How is the LCM calculated?
The most efficient method uses the identity lcm(a, b) = (a × b) ⁄ gcd(a, b) — the product of the two numbers divided by their greatest common divisor, computed via the fast Euclidean algorithm rather than listing multiples one at a time.
Why is the LCM useful for adding fractions?
It's the smallest common denominator two fractions can share, meaning the resulting sum requires no further simplification afterward. A larger common denominator, such as the plain product of the two original denominators, also works but usually leaves an unreduced fraction behind.
What is the LCM of two coprime numbers?
Simply their product. When two numbers share no common factor beyond 1 (their GCD is 1), the identity linking GCD and LCM forces their LCM to equal a × b exactly, since nothing smaller could be a multiple of both.
How is LCM related to GCD?
For any two positive whole numbers, gcd(a, b) × lcm(a, b) = a × b — the product of the two numbers always equals their GCD multiplied by their LCM, a useful identity for computing either one once the other is known.