Least common multiple (LCM)

Before you rely on it

What to check

Both original numbers must divide the result exactly, and that no smaller number does.

The common error

Where people go wrong with least common multiple (LCM)

Multiplying the two numbers together and calling it the LCM. That is a common multiple, it is often not the least one, and it is only correct when the two share no factors.

Worked through

Least common multiple (LCM) in plain numbers

The LCM of 4, 6 and 10 is 60. List multiples: 4, 8, 12, ... 60. Six: 6, 12, ... 60. Ten: 10, 20, ... 60. Sixty is the first number in all three lists; use prime factors for larger numbers: 4 = 2², 6 = 2 × 3, 10 = 2 × 5, so take the highest power of each prime: 2² × 3 × 5 = 60. In practice: buses leave every 12 and 18 minutes. They leave together every 36 minutes, the LCM. And to add 1/4 + 1/6 you convert to a common denominator of 12: 3/12 + 2/12 = 5/12.

Easily confused

Least common multiple (LCM) and its neighbours

The LCM is not the greatest common factor, but the two are partners; the GCF is the largest number that divides both, and the LCM is the smallest number that both divide into. For two numbers, their product equals the GCF times the LCM. If the numbers are coprime, the LCM is just their product, and the LCM is also called the lowest common multiple. It is different from the least common denominator, which is the LCM applied to denominators of fractions.

In real decisions

Where least common multiple (LCM) turns up

Adding and subtracting fractions with different denominators relies on it. Scheduling problems use it to find when repeating events align: gear teeth meeting again, shifts rotating together or traffic lights syncing. Music uses it for rhythms in different meters. Programmers use it to synchronize loop cycles. In manufacturing, packing sizes that divide evenly into a shipment come from common multiples.

Know the edges

Where least common multiple (LCM) stops being reliable

LCMs of several or large numbers can grow huge, faster than the numbers themselves. Calculators may overflow. The concept applies to positive integers, and conventions for zero and negatives vary. For fractions, using the LCM of denominators gives the smallest common denominator, but any common multiple works, just with larger numbers to simplify afterward.

Quick answers

What people ask about least common multiple (LCM)

How do I find the LCM using the GCF? Multiply the two numbers and divide by their GCF. For 12 and 18, the product is 216 and the GCF is 6, so the LCM is 36.

When do I need the LCM for fractions? When adding or subtracting fractions with different denominators. The LCM of the denominators is the smallest common denominator, which keeps numbers small.

Is the LCM ever smaller than the numbers? Never. It is at least as large as the largest number. It equals it only when the others divide it evenly.

What is the LCM of two primes? Their product. Two different primes share no factors, so the smallest number both divide is the product of the two.

How do I find the LCM of three numbers? Find the LCM of the first two, then find the LCM of that result with the third. Prime factors work too: take the highest power of each prime.