Grade by Grade
B. Number · Operations: Divisibility, Decimals, Fractions, Ratios and Percents

Prime Numbers, Composite Numbers and Factor Trees

B2.6 B2.2About 40 minutes
Learning goals
  • I can tell if a number is prime or composite.
  • I can make a factor tree to break a composite number into prime factors.
  • I can write a composite number as a product of its prime factors.

Words to know

prime number
A whole number greater than 1 with exactly two factors: 1 and itself. Examples: 2, 3, 5, 7, 11.
composite number
A whole number greater than 1 with more than two factors. Example: 6 has factors 1, 2, 3 and 6.
prime factor
A factor that is a prime number. The prime factors of 12 are 2 and 3.
factor tree
A diagram that breaks a number into factors, again and again, until every branch ends in a prime number.
prime factorization
A number written as a product of prime numbers only, like 60 = 2 × 2 × 3 × 5.

Let's start with arrays

Try to arrange 12 chairs in rows with the same number in each row. You can make 1 row of 12, 2 rows of 6, 3 rows of 4, 4 rows of 3, 6 rows of 2 or 12 rows of 1. Lots of ways!

Now try 7 chairs. You can only make 1 row of 7, or 7 rows of 1. That's it.

Numbers like 7 are special. They are called prime numbers.

12 chairs vs 7 chairs3 rows of 42 rows of 61 row of 12 (and the flips)12: lots of ways, so composite1 row of 72 rows? 1 left over ✗7: only 1 × 7, so prime
12 chairs make many rectangles, so 12 is composite. 7 chairs make only one row of 7: any other try leaves chairs over, so 7 is prime.

Prime or composite?

  • A prime number has exactly two factors: 1 and the number itself. 7 is prime (1 × 7).
  • A composite number has more than two factors. 12 is composite (1, 2, 3, 4, 6, 12).
  • 0 and 1 are neither prime nor composite. 1 has only one factor (itself).

The prime numbers less than 50 are:

2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47

2 is the only even prime. Every other even number is divisible by 2, so it has more than two factors.

Use your divisibility rules to spot composite numbers fast. 51 looks prime, but 5 + 1 = 6, so it's divisible by 3 (51 = 3 × 17). It is composite.

Factor trees

Every composite number can be written as a product of prime numbers. A factor tree helps you find them.

  1. Write the number at the top.
  2. Split it into any two factors (not 1). Draw two branches.
  3. If a factor is prime, circle it. That branch stops.
  4. If a factor is composite, split it again.
  5. When every branch ends in a circled prime, multiply the circled numbers. That is the prime factorization.
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A factor tree for 60. 60 = 6 × 10, then 6 = 2 × 3 and 10 = 2 × 5. The circled numbers are prime, so 60 = 2 × 2 × 3 × 5.
Worked example

Example 1: Two different trees for 60

Make a different factor tree for 60 than the one in the picture.

  1. Start with 60 = 4 × 15.
  2. 4 is composite: 4 = 2 × 2. Circle both 2s.
  3. 15 is composite: 15 = 3 × 5. Circle the 3 and the 5.
  4. Circled primes: 2, 2, 3, 5. Write them from smallest to largest.

60 = 2 × 2 × 3 × 5. The same primes as the first tree! No matter how you start, a number always has the same prime factorization.

Worked example

Example 2: Prime factorization of 72

Write 72 as a product of prime factors.

  1. Start with 72 = 8 × 9.
  2. 8 = 2 × 4, and 4 = 2 × 2. So 8 = 2 × 2 × 2.
  3. 9 = 3 × 3.
  4. Check: 2 × 2 × 2 × 3 × 3 = 8 × 9 = 72. ✓

72 = 2 × 2 × 2 × 3 × 3

Worked example

Example 3: Use divisibility rules to start

Write 126 as a product of prime factors.

  1. 126 is even, so start with 2: 126 = 2 × 63. Circle the 2.
  2. 63: the digit sum is 6 + 3 = 9, so 63 is divisible by 9. 63 = 7 × 9. Circle the 7.
  3. 9 = 3 × 3. Circle both.

126 = 2 × 3 × 3 × 7

Where to start

When you're stuck finding a factor, try the small primes in order: Is it divisible by 2? By 3? By 5? By 7? Your divisibility rules make this fast.

Just for fun

Funny but true: the biggest prime numbers found so far have more than 40 million digits. Writing one out by hand would take years, and you would need a very, very long piece of paper.

Hands-on activity

Prime hunt on a hundred chart

You need: a hundred chart (draw a 10 by 10 grid and write 1 to 100, or print one) and two colouring pencils.

This method is called the Sieve of Eratosthenes, named after a mathematician from ancient Greece.

  1. Cross out 1 (it is not prime).
  2. Circle 2. Then cross out all the other multiples of 2 (4, 6, 8, …).
  3. Circle 3, the next number not crossed out. Cross out every multiple of 3 that is left.
  4. Circle 5, then cross out its multiples. Then circle 7 and cross out its multiples.
  5. Every number left that isn't crossed out is prime. Circle them all.

Count them. You should find 25 primes from 1 to 100.

Math talk: "Why didn't you need to cross out multiples of 4 or 6?" (They were already crossed out as multiples of 2 or 3.)

Your turn

1

Is 29 prime or composite?

2

Is 39 prime or composite?

3

Write 24 as a product of prime factors.

4

Write 45 as a product of prime factors.

5

Write 50 as a product of prime factors.

6

Write 84 as a product of prime factors.

7

Which number has the prime factorization 2 × 3 × 7?

More practice: warm-up

1

List all the factors of 12.

2

Is 2 prime?

3

Is 1 prime?

4

Write 15 as a product of primes.

5

Is 21 prime or composite?

More practice: on your own

1

Write 90 as a product of prime factors.

2

Write 100 as a product of prime factors.

3

Write 96 as a product of prime factors.

4

Is 51 prime or composite?

5

What number is 2 × 2 × 3 × 3 × 5?

Challenge

1

Write 210 as a product of prime factors.

2

Liam's factor tree for 60 ends with 4, 3 and 5. He writes 60 = 4 × 3 × 5. What is wrong?

3

Which numbers less than 30 are the product of two equal primes?

Check your understanding

Quiz

1.

Which number is prime?

2.

What is the prime factorization of 36?

3.

Why is 1 not a prime number?

4.

What number is 2 × 2 × 5 × 7?

5.

Which is a correct first split for a factor tree of 75?

6.

How many prime factors does 48 have when you count repeats? (48 = 2 × 2 × 2 × 2 × 3)

For parents and teachers

Big idea: B2.6 asks Grade 6 students to represent composite numbers as a product of their prime factors, including with factor trees. Students do not need exponents (2³) in Grade 6; writing 2 × 2 × 2 is fine. We tagged B2.2 too, because divisibility rules are the fastest way to find a first factor.

Common difficulties:

  • Calling 1 prime, or thinking all odd numbers are prime (9, 15, 21, 27 are not).
  • Stopping too early with a composite number at the end of a branch (like 4 or 9).
  • Forgetting a repeated factor when writing the final answer. Count the circles.

How to help: Do the hundred-chart sieve together. Then ask your child to make two different factor trees for the same number (try 48 or 72) and notice that the primes at the end are always the same.