Greatest Common Factors and Least Common Multiples | Grade 6 Math | LevelKite

You already know that a factor of a number is a whole number that divides it with no remainder. For example, $3$ is a factor of $6$ because $6 \div 3 = 2$ with nothing left over.

Now you will compare two numbers at a time. Every number in this lesson is a whole number greater than $0$ .

A common factor of two numbers is a factor of both. The greatest common factor, or GCF, is the largest common factor. It is the biggest number that divides both numbers with no remainder.

To find the GCF of $6$ and $9$ , list every factor of each number. Then look for the numbers that are in both lists. A number in only one list, such as $2$ , is not a common factor.

Factors of 6 and 9

List	Numbers in the list
Factors of $6$	$1, 2, 3, 6$
Factors of $9$	$1, 3, 9$
Common factors	$1, 3$
Greatest common factor	$3$

Worked example

Find a GCF by listing factors

Find the greatest common factor of $24$ and $36$ .

List every factor of 24. Use factor pairs: each factor has a partner that multiplies with it to make 24, so the pairs give you every factor. The pairs are 1 and 24, 2 and 12, 3 and 8, and 4 and 6. Write the numbers in order.
$\text{Factors of 24: } 1, 2, 3, 4, 6, 8, 12, 24$
Do the same for 36. The pairs are 1 and 36, 2 and 18, 3 and 12, 4 and 9, and 6 and 6. Write 6 only once.
$\text{Factors of 36: } 1, 2, 3, 4, 6, 9, 12, 18, 36$
Pick out the numbers that are in both lists. These are the common factors, because each one divides 24 and also divides 36.
$\text{Common factors: } 1, 2, 3, 4, 6, 12$
The GCF is the largest common factor, so choose 12. Check it: 12 times 2 is 24 and 12 times 3 is 36, so 12 divides both numbers with no remainder.
$24 \div 12 = 2 \qquad 36 \div 12 = 3$

Answer

The greatest common factor of $24$ and $36$ is $12$ .

A multiple of a number is what you get when you multiply it by a counting number ( $1, 2, 3, \ldots$ ). The multiples of $6$ are $6, 12, 18, 24$ , and so on. They are the numbers you say when you count by sixes, and the list never ends.

A common multiple of two numbers is a multiple of both. The least common multiple, or LCM, is the smallest common multiple. It is the first number that both counts land on.

To find the LCM of $6$ and $9$ , count by sixes and count by nines. Then look for the first number that is in both lists. A number in only one list, such as $12$ , is not a common multiple.

Multiples of 6 and 9

List	Numbers in the list
Multiples of $6$	$6, 12, 18, 24, 30, 36, \ldots$
Multiples of $9$	$9, 18, 27, 36, \ldots$
Common multiples	$18, 36, \ldots$
Least common multiple	$18$

Worked example

Find an LCM by listing multiples

Find the least common multiple of $8$ and $10$ .

List multiples of the larger number, 10. Every common multiple of 8 and 10 is a multiple of 10, so the LCM must be in this list. Starting with the larger number means fewer numbers to test.
$10, 20, 30, 40, 50, \ldots$
Test the multiples in order, smallest first. Divide each one by 8. A remainder (R) means the number is not a multiple of 8.
$10 \div 8 = 1 \text{ R } 2 \qquad 20 \div 8 = 2 \text{ R } 4 \qquad 30 \div 8 = 3 \text{ R } 6$
Test the next multiple, 40. It divides by 8 with no remainder, so 40 is a multiple of 8 as well as a multiple of 10.
$40 \div 8 = 5 \qquad 40 \div 10 = 4$
Stop here. You tested from smallest to largest, so the first number that works is the least common multiple.

Answer

The least common multiple of $8$ and $10$ is $40$ .

Key concept

Two questions about the same two numbers

For two whole numbers greater than $0$ :

The greatest common factor (GCF) is the largest whole number that divides both numbers with no remainder.
The least common multiple (LCM) is the smallest number that is a multiple of both numbers.

A common factor goes into both numbers. Both numbers go into a common multiple. For $6$ and $9$ , the GCF is $3$ and the LCM is $18$ .

You already know how to write a prime factorization, which shows a number as a product of prime numbers (primes for short). You can use it to find both the GCF and the LCM of two whole numbers greater than $1$ . Those are the numbers that have a prime factorization.

Write the prime factorization of each number. Then match primes that are the same. Pair one prime from the first number with one from the second, and use each prime in only one pair. A prime left without a partner is unmatched.

GCF: a common factor must fit inside both factorizations, so you can build it only from matched pairs. Use them all: multiply one prime from each pair. If no primes match, the only common factor is $1$ , so the GCF is $1$ .
LCM: a common multiple must contain the whole prime factorization of each number. A matched pair needs only one copy, because that copy counts for both numbers. So multiply one prime from each pair and every unmatched prime.

Try this on $6$ and $9$ , where you already know the answers. Write $6 = 2 \times 3$ and $9 = 3 \times 3$ . The $3$ in $6$ pairs with one $3$ in $9$ . The $2$ in $6$ and the other $3$ in $9$ are unmatched.

GCF: there is one matched pair, so the GCF is $3$ .
LCM: multiply one $3$ from the pair by the unmatched $2$ and $3$ . You get $3 \times 2 \times 3 = 6 \times 3 = 18$ .

Both answers agree with your lists.

Worked example

Find the GCF and the LCM with primes

Use prime factorization to find the greatest common factor and the least common multiple of $8$ and $12$ .

Write the prime factorization of each number so that you can compare their primes.
$8 = 2 \times 2 \times 2 \qquad 12 = 2 \times 2 \times 3$
Match equal primes, one from each number. Two 2s from 8 pair with the two 2s from 12. Put the matched part in parentheses. Regrouping factors does not change a product, so both equations stay true. One 2 from 8 and the 3 from 12 are unmatched.
$8 = (2 \times 2) \times 2 \qquad 12 = (2 \times 2) \times 3$
Multiply the matched primes, one from each pair, to get the GCF. These primes are in both numbers, so their product divides both: 8 is 4 times 2, and 12 is 4 times 3.
$2 \times 2 = 4 \qquad 8 = 4 \times 2 \qquad 12 = 4 \times 3$
For the LCM, multiply one prime from each pair and every unmatched prime. The product contains all of 8 and all of 12, with nothing extra.
$(2 \times 2) \times 2 \times 3 = 4 \times 2 \times 3 = 8 \times 3 = 24$
Check with lists. The factors of 8 are 1, 2, 4, 8 and the factors of 12 are 1, 2, 3, 4, 6, 12, so the greatest factor they share is 4. The first multiple of 12 is 12, which is not a multiple of 8. The second is 24, and 8 divides it with no remainder.
$12 \div 8 = 1 \text{ R } 4 \qquad 24 \div 8 = 3$

Answer

The greatest common factor of $8$ and $12$ is $4$ , and their least common multiple is $24$ .

Takeaway. The GCF and the LCM answer two different questions about the same two numbers. The GCF is the largest number that divides both. The LCM is the smallest number that is a multiple of both. You can find each one with lists or by matching primes.

Next, you will use the GCF and the distributive property to factor a sum, solve word problems about the GCF and the LCM, and find mistakes in GCF and LCM work.