B. Number · Operations: Exponents, Factors, Fractions, Decimals, Integers, Percents and Proportions

The Greatest Common Factor

B2.6 A1About 40 minutes

Learning goals

I can find the greatest common factor (GCF) of two or three whole numbers up to 144.
I can use lists of factors or prime factorization to find the GCF.
I can use the GCF to simplify fractions and to solve sharing problems.

Words to know

factor
A whole number that divides into another number with no remainder. 6 is a factor of 18.

common factor
A number that is a factor of two or more numbers. 3 is a common factor of 12 and 18.

greatest common factor (GCF)
The largest number that is a factor of all the numbers. The GCF of 12 and 18 is 6.

prime factorization
A number written as a product of prime numbers, like 24 = 2³ × 3.

Remember from Grade 6

In Grade 6 you found factors, used divisibility rules, and broke numbers into prime factors with factor trees. In Grade 7 you will use those skills to find the greatest common factor of two or three numbers.

Let's start with snack bags

Your class is packing snack bags for a winter carnival. You have 24 granola bars and 36 apples. Every bag must be exactly the same, and nothing can be left over. What is the greatest number of bags you can make?

The number of bags must divide 24 and 36. It must be a common factor. The largest one is 12. So you can make 12 bags, each with 24 ÷ 12 = 2 granola bars and 36 ÷ 12 = 3 apples.

A pile of granola bars in silver wrappers, a bowl of red apples, and a row of brown paper snack bags on a classroom table. — To fill identical snack bags with nothing left over, the number of bags must be a common factor of the number of bars and apples. AI-generated image

With the GCF of 24 and 36, which is 12, you get 12 identical bags: 12 × 2 = 24 bars and 12 × 3 = 36 apples, with nothing left over.

Method 1: List the factors

List all the factors of each number in pairs, so you don't miss any. Then circle the ones they share.

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. The greatest is 12.

Method 2: Prime factorization

Write each number as a product of primes. The GCF is the product of the primes they share.

24 = 2 × 2 × 2 × 3
36 = 2 × 2 × 3 × 3

They share 2, 2 and 3. GCF = 2 × 2 × 3 = 12.

A Venn diagram of prime factors. The overlap holds the primes 24 and 36 share: 2, 2 and 3. Multiply them: GCF = 2 × 2 × 3 = 12.

Worked example

Example 1: Two larger numbers

Find the GCF of 96 and 144.

Prime factorization: 96 = 2⁵ × 3 (2 × 2 × 2 × 2 × 2 × 3).
144 = 2⁴ × 3² (2 × 2 × 2 × 2 × 3 × 3).
They share four 2s and one 3: 2⁴ × 3 = 16 × 3 = 48.
Check: 96 ÷ 48 = 2 and 144 ÷ 48 = 3. ✓

GCF(96, 144) = 48

Worked example

Example 2: Three numbers

Find the GCF of 18, 30 and 42.

18 = 2 × 3 × 3, 30 = 2 × 3 × 5, 42 = 2 × 3 × 7.
All three share one 2 and one 3.
GCF = 2 × 3 = 6.

GCF(18, 30, 42) = 6

Worked example

Example 3: Simplify a fraction

Write 36/60 in lowest terms.

The GCF of 36 and 60 is 12. (36 = 2² × 3², 60 = 2² × 3 × 5. They share 2² × 3 = 12.)
Divide the numerator and the denominator by 12: 36 ÷ 12 = 3 and 60 ÷ 12 = 5.

36/60 = 3/5 in one step.

Quick checks

The GCF can never be bigger than the smallest number.
If the smaller number divides the bigger one, the smaller number is the GCF: GCF(7, 21) = 7.
If two numbers share no prime factors, their GCF is 1: GCF(9, 16) = 1.

Hands-on activity

Pasta packs

You need: two kinds of dry pasta or buttons (for example 16 macaroni and 24 shells), a few small bowls or paper cups.

Count out 16 of one kind and 24 of the other.
Share them into bowls so that every bowl is the same and nothing is left over.
Try 2 bowls, 4 bowls, 8 bowls... What is the greatest number of bowls that works?
Check with the factor lists and with prime factors.
Try again with 18 and 27.

What to look for: 16 and 24 → 8 bowls (2 macaroni and 3 shells each). 18 and 27 → 9 bowls (2 and 3 each).

Math talk: "Why doesn't 6 bowls work for 16 and 24?"

Just for fun

Why do 24 and 36 get along so well?

They have so much in common: 1, 2, 3, 4, 6 and 12!

Your turn

Find the GCF of 12 and 18.

Find the GCF of 15 and 25.

Find the GCF of 16 and 40.

Find the GCF of 14 and 15.

Find the GCF of 36 and 60.

Find the GCF of 27, 45 and 63.

Use the GCF to write 28/42 in lowest terms.

More practice: warm-up

List all the factors of 20.

Find the GCF of 8 and 12.

Find the GCF of 10 and 25.

Find the GCF of 7 and 21.

Find the GCF of 9 and 16.

More practice: on your own

Find the GCF of 48 and 72.

Find the GCF of 84 and 126.

Two ribbons are 64 cm and 80 cm long. You cut both into pieces that are all the same length, as long as possible, with nothing left over. How long is each piece? How many pieces do you get?

Use the GCF to write 54/90 in lowest terms.

Find the GCF of 30, 45 and 75.

Challenge

A floor is 120 cm by 144 cm. You want to cover it with the largest square tiles possible, with no cutting. What size are the tiles, and how many do you need?

Find two numbers between 50 and 100 whose GCF is 12.

Can the GCF of two numbers ever be greater than the smaller number? Explain.

Check your understanding

Quiz

What is the GCF of 18 and 24?

What is the GCF of 9 and 20?

Which problem is solved with the GCF?

If a = 2² × 3 and b = 2 × 3², what is the GCF of a and b?

What is the GCF of 32 and 48?

Write 24/32 in lowest terms.

For parents and teachers

Big idea: The Ontario curriculum (B2.6) asks Grade 7 students to determine the greatest common factor for a variety of whole numbers up to 144, as well as the lowest common multiple (next lesson). The GCF is used to simplify fractions and to make equal groups with nothing left over.

Common difficulties:

Confusing GCF with LCM. The GCF is never larger than the smallest number; the LCM is never smaller than the largest.
Missing factors in a list. Listing factors in pairs (1 × 24, 2 × 12, 3 × 8, 4 × 6) prevents this.
Multiplying all the primes instead of only the shared ones.

How to help: Do the pasta packs activity and talk through why some numbers of bowls don't work. This lesson builds A1 skills: comparing strategies and choosing the most efficient one.