What is the least common multiple and the greatest common factor, and how do you find them?

The least common multiple (LCM) is the smallest number that two numbers both divide into, and the greatest common factor (GCF) is the largest number that divides both. For 12 and 18, the GCF is 6 and the LCM is 36. You find them by listing factors and multiples, by prime factorization, or with the ladder method.

What is the least common multiple of 6 and 8?

The least common multiple of 6 and 8 is 24. The multiples of 6 are 6, 12, 18, 24 and 30, and the multiples of 8 are 8, 16, 24 and 32, so 24 is the first number on both lists. Multiplying 6 × 8 gives 48, which is a common multiple but not the least one.

What is the difference between GCF and LCM?

The difference between GCF and LCM is direction: the greatest common factor is the largest number that divides both numbers, so it is never larger than the smaller number. The least common multiple is the smallest number both numbers divide into, so it is never smaller than the larger number. For 12 and 18, the GCF is 6 and the LCM is 36.

How do you find the LCM quickly?

The quickest way to find the LCM of two numbers is to multiply them and divide by their greatest common factor. For 6 and 8, the GCF is 2, so the LCM is 6 × 8 ÷ 2 = 48 ÷ 2 = 24. For small numbers, listing the multiples of the larger number until one is divisible by the smaller number is also fast.

Is GCF the same as GCD and HCF?

The greatest common factor (GCF), the greatest common divisor (GCD) and the highest common factor (HCF) are three names for the same number. American schools say GCF, mathematicians often say GCD, and schools in the UK say HCF. The UK also calls the least common multiple the lowest common multiple, with the same abbreviation, LCM.

Least Common Multiple and Greatest Common Factor

A fraction homework sheet asks for the least common multiple of 6 and 8, and the next line asks for the greatest common factor of 12 and 18. The two terms sound alike and children mix them up. Both rest on the prime factors of each number, and three short methods find them.

What are the greatest common factor and the least common multiple?

The greatest common factor (GCF) of two numbers is the largest number that divides both evenly, and the least common multiple (LCM) is the smallest number that both divide into. For 12 and 18, the GCF is 6 and the LCM is 36.

The factors of 12 are 1, 2, 3, 4, 6 and 12, and the factors of 18 are 1, 2, 3, 6, 9 and 18. The common factors are 1, 2, 3 and 6, so the greatest is 6.

The multiples of 12 are 12, 24, 36, 48, and the multiples of 18 are 18, 36, 54. The first number on both lists is 36, so the least common multiple is 36.

Factors and multiples: the two words to get straight

A factor of a number divides it with no remainder, so the factors of a number are never larger than the number itself. A multiple is the number times 1, 2, 3 and so on, so the multiples are never smaller than the number and the list never ends.

Common Core standard 4.OA.B.4 sets up this vocabulary in grade 4, where students "find all factor pairs for a whole number in the range 1–100" and "recognize that a whole number is a multiple of each of its factors". The factor pairs of 18 are 1 × 18, 2 × 9 and 3 × 6, and every number in those pairs is a factor of 18.

A child who keeps "factor" and "multiple" apart has most of the GCF and LCM problem solved. The factor pairs game drills the first word and the multiples game drills the second.

Factor trees break 12 into 2, 2 and 3 and 18 into 2, 3 and 3; a Venn diagram puts the shared 2 and 3 in the overlap, giving 2 × 3 = 6 and 2 × 2 × 3 × 3 = 36.

Why GCF and LCM matter in grades 4 to 6

The greatest common factor and least common multiple become a named skill in grade 6. Common Core standard 6.NS.B.4 asks students to "find the greatest common factor of two whole numbers less than or equal to 100 and the least common multiple of two whole numbers less than or equal to 12".

The same standard links the GCF to the distributive property, with the example "express 36 + 8 as 4 (9 + 2)". The GCF of 36 and 8 is 4, so 36 + 8 = 4 × 9 + 4 × 2 = 4 × (9 + 2) = 44.

England spreads the skill over several years. The national curriculum for mathematics in England asks Year 6 pupils to "identify common factors, common multiples and prime numbers" and names the "highest common factor" and "lowest common multiple" in key stage 3.

When to use the GCF and when to use the LCM

The GCF answers "how many equal groups can I make", and the LCM answers "when do two counts line up". A child who asks which one a word problem needs can test both questions against it.

Equal groups (GCF): 18 apples and 24 oranges go into identical bags with nothing left over. The GCF of 18 and 24 is 6, so you get 6 bags, each with 3 apples and 4 oranges.
Simplifying fractions (GCF): Dividing the top and bottom of 18/24 by their GCF, 6, gives 3/4 in one step. The guide to fractions explained for parents covers equivalent fractions in more depth.
Meeting again (LCM): One bus leaves every 6 minutes and another every 8 minutes, both at 9:00. The LCM of 6 and 8 is 24, so they next leave together at 9:24.
Common denominators (LCM): To add 5/6 + 3/8, use the LCM of 6 and 8 as the denominator: 5/6 = 20/24 and 3/8 = 9/24, so the sum is 29/24.

Play free in your browserMultiples RushGrade 3–5

How to find the least common multiple and greatest common factor, step by step

Prime factorization finds the least common multiple and greatest common factor in one pass, and it works for any pair of numbers. The steps below use 24 and 36.

Break each number into primes. Build a factor tree or divide by the smallest prime that fits: 24 = 2 × 2 × 2 × 3 and 36 = 2 × 2 × 3 × 3. The guide to prime numbers explained shows factor trees step by step.
Write the factors with powers. Group the repeats: 24 = 2³ × 3 and 36 = 2² × 3². The powers make the next two steps a matter of comparing small numbers.
Find the GCF from the shared primes. Take only the primes both numbers have, each with its lower power: 2² × 3 = 4 × 3 = 12.
Find the LCM from all the primes. Take every prime that appears in either number, each with its higher power: 2³ × 3² = 8 × 9 = 72.
Check both answers. The GCF must divide both numbers (24 ÷ 12 = 2, 36 ÷ 12 = 3), and both numbers must divide the LCM (72 ÷ 24 = 3, 72 ÷ 36 = 2).

The OpenStax Prealgebra chapter on prime factorization and the least common multiple sets out the same method, with shared primes counted once.

Play free in your browserFactor PairsGrade 3–5

The part people get wrong: lowest powers vs highest powers

The rule for the powers is the step children swap most often. The GCF takes the lowest power of each shared prime, because a common factor cannot use more 2s than the number with fewer 2s has.

The LCM takes the highest power of every prime, because a common multiple has to contain all the 2s of the number with the most 2s. For 24 and 36, 24 has three 2s, so any multiple of 24 needs 2³, and 36 has two 3s, so any multiple of 36 needs 3².

Listing vs prime factorization vs the ladder method

The listing method, prime factorization and the ladder method give the same GCF and LCM and differ in speed.

Method	How it works	Best for	Limits
Listing	Write the factors (for the GCF) or the multiples (for the LCM) of each number and pick the first match	Small numbers, grades 4–5, first lessons	Slow above 50; easy to stop the list too early
Prime factorization	Factor tree for each number; GCF = shared primes with lowest powers, LCM = all primes with highest powers	Grade 6 and up, three or more numbers, numbers up to 100 and beyond	Needs fluent division by 2, 3, 5 and 7
Ladder (cake) method	Divide both numbers by a common prime, row after row, until no common factor is left	Finding the GCF and LCM together, fast, on paper	Works on two numbers at a time; the reading-off step is easy to forget

The ladder method, worked

The ladder method divides both numbers by a prime they share, writes the quotients underneath, and repeats until the bottom pair has no common factor except 1. For 24 and 36, divide by 2 to get 12 and 18, by 2 again to get 6 and 9, and by 3 to get 2 and 3.

The divisors down the left side multiply to the GCF: 2 × 2 × 3 = 12. The LCM is the GCF times the numbers left on the bottom row: 12 × 2 × 3 = 72.

The ladder method for 24 and 36: dividing by 2, 2 and 3 leaves 2 and 3 on the bottom row, so the GCF is 2 × 2 × 3 = 12 and the LCM is 12 × 2 × 3 = 72, and 12 × 72 = 24 × 36 = 864.

The GCF × LCM shortcut

The product of the greatest common factor and the least common multiple of two numbers always equals the product of the two numbers. For 12 and 18, the GCF is 6 and the LCM is 36, and 6 × 36 = 216 = 12 × 18.

The shortcut turns one answer into the other. Once you know the GCF, the LCM is a × b ÷ GCF: for 6 and 8 the GCF is 2, so the LCM is 48 ÷ 2 = 24.

For small numbers, count up the multiples of the larger number until the smaller one divides it: 8, 16, 24, and 6 divides 24. The guide on how to learn the times tables fast builds the recall this needs.

Why the shortcut works

The shortcut works because the GCF and the LCM share out the prime factors of the two numbers between them. For 12 = 2 × 2 × 3 and 18 = 2 × 3 × 3, the GCF takes the shared 2 and 3, and the LCM takes the shared 2 and 3 again plus the extra 2 and 3.

The primes of 6 × 36 are 2, 3 and 2, 2, 3, 3, the same six primes as in 12 × 18.

The rule holds for two numbers only. For three numbers such as 4, 6 and 8, the GCF is 2 and the LCM is 24, and 2 × 24 = 48 is far from 4 × 6 × 8 = 192.

Common GCF and LCM mistakes

Four mistakes cause most of the wrong answers on a GCF and LCM worksheet, and each has a one-line fix.

Mixing up the two: A GCF can never be larger than the smaller number, and an LCM can never be smaller than the larger number. An answer of 36 for the GCF of 12 and 18 fails that check at once.
Assuming the LCM is always a × b: Multiplying gives a common multiple, but the least one only when the GCF is 1, as with 4 and 9 (LCM 36). For 6 and 8, 6 × 8 = 48 while the LCM is 24.
Stopping the list too early: Listing 9, 18, 27 and 12, 24 shows no match, which tempts a child to answer 9 × 12 = 108. One more step on each list reaches 36, the LCM.
Treating 1 as a prime: The number 1 is not prime, so it never appears in a factor tree. Writing 12 = 1 × 12 gets the tree nowhere, while 12 = 2 × 6 starts it.

Two edge cases come up in grade 6. When one number divides the other, as with 4 and 12, the GCF is the smaller number (4) and the LCM is the larger (12). When two numbers share no prime, as with 8 and 15, the GCF is 1 and the LCM is 8 × 15 = 120.

How to check your child has it

Five quick questions show whether the GCF and LCM have landed, and each takes under a minute.

The definitions: Ask what the GCF and the LCM of two numbers are, with no example. Both answers need the word "both".
The famous pair: Ask for the LCM of 6 and 8. The answer is 24, and an answer of 48 shows the a × b habit.
A word problem sorted: Ask whether "18 apples and 24 oranges in identical bags" needs the GCF or the LCM. The answer is the GCF, 6 bags.
Prime factorization: Ask for the GCF and the LCM of 24 and 36 by factor trees or the ladder. The answers are 12 and 72.
The check: Ask your child to prove the answers by dividing: 12 divides 24 and 36, and 24 and 36 both divide 72.

The missing factor game builds the division recall behind every one of these checks.

Your GCF and LCM checklist

Say the definitions: the GCF divides both numbers, and both numbers divide the LCM.
Size-check the answer: the GCF is at most the smaller number, and the LCM is at least the larger one.
Pick the method by size: listing below 20, prime factorization or the ladder above it.
Use the shortcut: LCM = a × b ÷ GCF for two numbers.
Match the word problem: equal groups and simplifying use the GCF; meeting times and common denominators use the LCM.

The same skills appear across the grade pages for math games for 4th graders and math games for 6th graders, which list the factor and multiple games by grade.

MathIt's free iOS and Android app has about 90 game types with adaptive difficulty, so factor and multiple practice gets harder only after the easier pairs come out right, and 22 of the games play free in a browser with no sign-up.