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GCF LCM

Find GCF and LCM

Master the Greatest Common Factor and Least Common Multiple: the two essential tools for simplifying fractions, finding common denominators, and rewriting expressions.

6th Grade Math

What Is a Factor?

A factor of a number is any whole number that divides into it evenly, with no remainder. For example, the factors of 12 are all the whole numbers that divide 12 exactly.

Factors of 12 1 2 3 4 6 12 Each divides 12 with zero remainder

What Is a Multiple?

A multiple of a number is the result of multiplying that number by any whole number. Multiples go on forever: the list never stops.

First multiples of 4 4 8 12 16 20 24 ... \(4\times1\) \(4\times2\) \(4\times3\) \(4\times4\) \(4\times5\) \(4\times6\) Multiples are unlimited: just keep multiplying!

The Greatest Common Factor

The Greatest Common Factor (GCF) of two numbers is the largest factor they share. It tells you how much the two numbers have in common.

Factors of 24 1 2 3 4 6 8 12 24 Factors of 36 1 2 3 4 6 9 12 18 36 Common factors: 1, 2, 3, 4, 6, 12 GCF = 12

The Least Common Multiple

The Least Common Multiple (LCM) of two numbers is the smallest number that both numbers divide into evenly. It is the first place their multiples overlap.

Multiples of 4 4 8 12 16 20 24 28 32 ... Multiples of 6 6 12 18 24 30 36 ... LCM = 12

GCF vs LCM: Key Differences

GCF
The largest factor two numbers share. Always less than or equal to the smaller number.
LCM
The smallest multiple two numbers share. Always greater than or equal to the larger number.
GCF LCM
Direction Work down from the numbers Work up from the numbers
Result size \(\leq\) smaller number \(\geq\) larger number
Example: 8, 12 4 24
Used for Simplifying fractions Common denominators

Finding GCF: List the Factors

1
List all factors of each number.
2
Circle the common factors: numbers that appear in both lists.
3
Pick the greatest one. That is the GCF.

Finding LCM: List the Multiples

1
List the first several multiples of each number.
2
Find the first match: the smallest number that appears in both lists.
3
That match is the LCM.

Example: GCF of \(36\) and \(48\)

Factors of 36 Factors of 48 1, 2, 3, 4, 6, 9, 12, 18, 36 1, 2, 3, 4, 6, 8, 12, 16, 24, 48 Common: 1, 2, 3, 4, 6, 12 appear in both lists GCF = 12

Example: LCM of \(6\) and \(8\)

Multiples of 6 6, 12, 18, 24, 30, 36, 42, 48 ... Multiples of 8 8, 16, 24, 32, 40, 48 ... LCM = 24

GCF and the Distributive Property

You can use the GCF to rewrite a sum of two numbers as a product. This connects GCF to the distributive property:

Express \(36 + 8\) using the GCF
\(\text{GCF}(36, 8) = 4\)
\(36 + 8 = 4(9 + 2)\)
1
Find GCF: \(\text{GCF}(36, 8) = \) 4
2
Divide each number by the GCF: \(36 \div 4 = 9\), \(8 \div 4 = 2\)
3
Write as product: \(4(9 + 2)\). Notice 9 and 2 share no common factor other than 1.

Practice: Rewriting Sums

\(15 + 25\)
GCF = 5
\(5(3 + 5)\)
\(24 + 18\)
GCF = 6
\(6(4 + 3)\)
\(56 + 72\)
GCF = 8
\(8(7 + 9)\)

Prime Factorization Method

For larger numbers, break each into its prime factors. Then use the primes to find both GCF and LCM at once.

60
\(2 \times 2 \times 3 \times 5\)
 2 2 3 5 48
\(2 \times 2 \times 2 \times 2 \times 3\)
 2 2 2 2 3 GCF: shared primes at lowest power
\(2^2 \times 3 = 12\)
 LCM: all primes at highest power
\(2^4 \times 3 \times 5 = 240\)
Verify: \(\text{GCF} \times \text{LCM} = 12 \times 240 = 2880\)
\(a \times b = 60 \times 48 = 2880\) ✓

Common Mistakes to Avoid

✕ Mixing Up GCF and LCM
GCF is the greatest shared factor (goes down). LCM is the least shared multiple (goes up). Remember: Factor = Fewer, Multiple = More.
✕ Stopping Too Early
Picking a common factor that isn't the greatest. Always check that no larger shared factor exists.
✕ Multiplying for LCM
Students often think \(\text{LCM}(a,b) = a \times b\). This only works when \(\text{GCF}(a,b) = 1\). Otherwise the LCM is smaller than the product.

Summary at a Glance

GCF LCM
Full name Greatest Common Factor Least Common Multiple
Method 1 List all factors, pick largest shared List multiples, pick smallest shared
Method 2 Prime factorization: lowest powers Prime factorization: highest powers
Relationship \(\text{GCF}(a,b) \times \text{LCM}(a,b) = a \times b\)
Distributive link \(a + b = \text{GCF}(a,b)\Big(\frac{a}{\text{GCF}} + \frac{b}{\text{GCF}}\Big)\)

Question 1 of 5

What is the GCF of \(24\) and \(36\)?

Question 2 of 5

What is the LCM of \(4\) and \(10\)?

Question 3 of 5

Which expression rewrites \(48 + 18\) using the distributive property with their GCF?

Question 4 of 5

What is the LCM of \(6\) and \(9\)?

Question 5 of 5

If \(\text{GCF}(a, b) = 5\) and \(a \times b = 150\), what is \(\text{LCM}(a, b)\)?