M6MATH-4.4

GCF, LCM & the Distributive Property

Learn to find the greatest common factor (GCF) and least common multiple (LCM) of whole numbers, and use the distributive property to factor expressions.

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What this lesson covers

The greatest common factor and least common multiple are two powerful tools that help you break down numbers and see their hidden structure. Once you can find them, you'll use the distributive property to rewrite expressions in ways that make math easier—a skill you'll build on for years. In this lesson, you'll discover how all three ideas work together to simplify problems and reveal patterns in numbers.

What is the Greatest Common Factor (GCF)?

The greatest common factor of two numbers is the largest number that divides evenly into both of them. For example, the factors of 24 are 1, 2, 3, 4, 6, 8, 12, and 24. The factors of 36 are 1, 2, 3, 4, 6, 9, 12, 18, and 36. The factors they share in common are 1, 2, 3, 4, 6, and 12. The greatest of these is 12, so GCF(24,36)=12\text{GCF}(24, 36) = 12.

There are two main ways to find the GCF. The first is to list all factors of each number and find the largest one they share. This works well for smaller numbers. The second way is to use prime factorization: write each number as a product of its prime factors, then multiply only the prime factors they have in common (using the lowest power of each). For instance, 24=23×324 = 2^3 \times 3 and 36=22×3236 = 2^2 \times 3^2. The common prime factors are 222^2 and 313^1, so GCF(24,36)=22×3=12\text{GCF}(24, 36) = 2^2 \times 3 = 12.

Finding the GCF is especially useful when you need to simplify fractions, factor expressions, or divide items into equal groups.

What is the Least Common Multiple (LCM)?

The least common multiple of two numbers is the smallest number that is a multiple of both of them. A multiple of a number is the result of multiplying it by any whole number. For example, the multiples of 6 are 6, 12, 18, 24, 30, 36, ... and the multiples of 8 are 8, 16, 24, 32, 40, ... The smallest number that appears in both lists is 24, so LCM(6,8)=24\text{LCM}(6, 8) = 24.

The easiest way to find the LCM is to list multiples of the larger number until you find one that is also a multiple of the smaller number. Another approach uses prime factorization: write each number as a product of primes, then multiply together each prime factor that appears, using the highest power that shows up in either factorization. For 6=2×36 = 2 \times 3 and 8=238 = 2^3, the LCM includes 232^3 (the highest power of 2) and 313^1 (which appears only in 6), giving LCM(6,8)=23×3=24\text{LCM}(6, 8) = 2^3 \times 3 = 24.

You'll use the LCM when adding or subtracting fractions (to find a common denominator) and when solving problems about repeating cycles or patterns.

The Distributive Property and Factoring

The distributive property says that a(b+c)=ab+aca(b + c) = ab + ac. This means multiplying a sum is the same as multiplying each part and adding the results. You've probably used it to expand expressions, but it also works backward to factor them.

Factoring is reversing the distributive property. If you have a sum like 12+1812 + 18, you can write it as 6(2)+6(3)=6(2+3)6(2) + 6(3) = 6(2 + 3) because 6 is a common factor of both 12 and 18. Notice that 6 is the GCF of 12 and 18. In general, if two terms share a common factor, you can pull that factor out in front using the distributive property.

For example: 15+25=5(3)+5(5)=5(3+5)=5(8)15 + 25 = 5(3) + 5(5) = 5(3 + 5) = 5(8). Here, 5 is the GCF of 15 and 25, and we factored the expression by writing 5 outside the parentheses. This form is often more useful because it shows the structure of the numbers and can make calculation simpler.

How GCF and LCM Connect to Factoring

The GCF is essential for factoring expressions correctly. When you factor out a common factor from a sum, you must factor out the greatest common factor to get the most simplified form. For instance, 24+3624 + 36 can be written as 12(2)+12(3)=12(2+3)=12(5)12(2) + 12(3) = 12(2 + 3) = 12(5). You could also write it as 6(4)+6(6)=6(10)6(4) + 6(6) = 6(10), but this is less simplified because you used 6 instead of the GCF, which is 12.

The LCM is less directly connected to factoring a single sum, but it matters when you're combining multiple expressions or solving word problems. For example, if you have 20+30+4020 + 30 + 40, the GCF of all three numbers is 10, so you can write it as 10(2)+10(3)+10(4)=10(2+3+4)=10(9)10(2) + 10(3) + 10(4) = 10(2 + 3 + 4) = 10(9).

A common mistake is forgetting to find the GCF first, or not checking whether your factored form is fully simplified. Always verify: divide each original term by the factor you pulled out. If the results share another common factor, you haven't used the GCF.

Real-World Applications

GCF and LCM appear in many practical situations. If you're organizing objects—say, arranging 24 cookies and 36 brownies into equal plates with no leftovers—the GCF tells you the maximum number of plates you can make (12 plates, each with 2 cookies and 3 brownies). If you need items to repeat on a schedule, like one task happening every 6 days and another every 8 days, the LCM tells you when they'll happen on the same day again: every 24 days. Factoring using the distributive property appears when you calculate areas of combined rectangles or when you simplify calculations by grouping like quantities together.

Key terms

Greatest Common Factor (GCF).
The largest number that divides evenly into two or more given numbers.
Least Common Multiple (LCM).
The smallest number that is a multiple of two or more given numbers.
Factor.
A number that divides evenly into another number with no remainder, or a number that is multiplied in a product.
Multiple.
The product of a whole number and any other whole number. For example, 15, 20, and 25 are multiples of 5.
Prime Factorization.
Writing a number as the product of all its prime factors. For example, 24=23×324 = 2^3 \times 3.
Distributive Property.
The rule a(b+c)=ab+aca(b + c) = ab + ac, which allows you to multiply a sum by distributing the multiplication across each term.
Factoring.
Using the distributive property in reverse to rewrite a sum as a product by pulling out a common factor.

Worked example

Express 36 + 48 using the distributive property by factoring out the GCF.
First, find the GCF of 36 and 48. List the factors of 36: 1, 2, 3, 4, 6, 9, 12, 18, 36. List the factors of 48: 1, 2, 3, 4, 6, 8, 12, 16, 24, 48. The common factors are 1, 2, 3, 4, 6, and 12. The greatest is 12.

Next, divide each term by the GCF: 36÷12=336 \div 12 = 3 and 48÷12=448 \div 12 = 4.

Now use the distributive property in reverse (factoring) to rewrite the sum:36+48=12(3)+12(4)=12(3+4)=12(7)36 + 48 = 12(3) + 12(4) = 12(3 + 4) = 12(7)You can verify this is correct by expanding: 12(7)=8412(7) = 84, and 36+48=8436 + 48 = 84. ✓

Practice questions

What is the greatest common factor of 20 and 50?
  1. 5
  2. 10
  3. 20
  4. 50

Answer: 10

To find the GCF of 20 and 50, list the factors of each. Factors of 20 are 1, 2, 4, 5, 10, 20. Factors of 50 are 1, 2, 5, 10, 25, 50. The common factors are 1, 2, 5, and 10. The greatest is 10. You can also use prime factorization: 20=22×520 = 2^2 \times 5 and 50=2×5250 = 2 \times 5^2. The common prime factors are 212^1 and 515^1, so the GCF is 2×5=102 \times 5 = 10.
What is the least common multiple of 9 and 12?
  1. 36
  2. 24
  3. 18
  4. 12

Answer: 36

Multiples of 9 are 9, 18, 27, 36, ... Multiples of 12 are 12, 24, 36, ... The smallest number that appears in both lists is 36. Using prime factorization: 9=329 = 3^2 and 12=22×312 = 2^2 \times 3. The LCM includes the highest power of each prime: 22×32=4×9=362^2 \times 3^2 = 4 \times 9 = 36.
Factor 28 + 42 using the distributive property by pulling out the GCF. What is the factored form?

Answer: 14(2+3)14(2 + 3) or 14(5)14(5)

The factors of 28 are 1, 2, 4, 7, 14, 28. The factors of 42 are 1, 2, 3, 6, 7, 14, 21, 42. The common factors are 1, 2, 7, and 14. The GCF is 14. Divide each term by 14: 28÷14=228 \div 14 = 2 and 42÷14=342 \div 14 = 3. So 28+42=14(2+3)=14(5)28 + 42 = 14(2 + 3) = 14(5). Both 14(2+3)14(2 + 3) and 14(5)14(5) are correct factored forms.

FAQ

What is the difference between GCF and LCM?
The GCF is the largest number that divides into both numbers, while the LCM is the smallest number that both numbers divide into. GCF is used for factoring and simplifying; LCM is used for finding common denominators and solving problems about cycles. For example, GCF(12, 18) = 6, but LCM(12, 18) = 36.
Can two numbers have more than one common multiple?
Yes! Any two numbers have infinitely many common multiples. The LCM is just the smallest one. For example, common multiples of 6 and 8 include 24, 48, 72, 96, and so on. They're all multiples of the LCM (24).
Why do we factor out the GCF and not just any common factor?
When you factor, you want the most simplified form. Factoring out the GCF makes the numbers inside the parentheses as small as possible and reveals the cleanest structure of the expression. If you factor out a smaller common factor, the remaining numbers still share a common factor, which means your factoring isn't complete.
How is the distributive property connected to finding the GCF?
The distributive property in reverse (factoring) lets you rewrite a sum by pulling out a common factor. To do this correctly and completely, that common factor should be the GCF. The GCF tells you the largest factor you can pull out, making a(b+c)=ab+aca(b + c) = ab + ac work with the greatest possible aa.

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