015. Prime Factorisation Using Factor Trees

Learning Intentions

• To understand that composite numbers can be broken down into a product of prime factors
• use a factor tree to find the prime factors of a number (including repeated factors)
• express a prime decomposition using powers of prime numbers

Pre-requisite Summary

• Understanding prime and composite numbers
• Ability to find factors of a number
• Knowledge of multiplication facts
• Understanding repeated factors
• Familiarity with powers (index notation)

Worked Examples

Worked Example 1

Break into prime factors:

a) $12$
b) $18$

Worked Example 2

Use a factor tree:

a) $24$
b) $30$

Worked Example 3

Use a factor tree with repeated factors:

a) $36$
b) $72$

Worked Example 4

Write prime factorisation using powers:

a) $24$
b) $90$

Problems

Problem 1

Break into prime factors:

a) $20$
b) $28$

Problem 2

Use a factor tree:

a) $32$
b) $45$

Problem 3

Use a factor tree with repeated factors:

a) $48$
b) $84$

Problem 4

Write prime factorisation using powers:

a) $40$
b) $108$

Exercises

Understanding and Fluency

1. Find prime factors:
   a) $16$
   b) $27$
   c) $50$

2. Use factor trees:
   a) $18$
   b) $42$
   c) $63$

3. Find prime factors (with repeats):
   a) $64$
   b) $54$
   c) $75$

4. Write using powers of primes:
   a) $24$
   b) $36$
   c) $48$

5. Write using powers of primes:
   a) $72$
   b) $90$
   c) $100$

6. Mixed practice:
   a) $60$
   b) $84$
   c) $120$

Reasoning

7. Explain why prime factorisation of a number is unique.

8. A student stops factorising $24=4\times 6$. Explain why this is incomplete.

9. Why must factor trees end only in prime numbers?

10. Explain why $36$ and $72$ share some prime factors.

Problem-solving

11. Find two different numbers with prime factorisation $2^2\times 3$.

12. A number has prime factors $2^3\times 5$. What is the number?

13. Find the prime factorisation of $96$ and write using powers.

14. Which number has prime factorisation $3^2\times 5$?

15. Find two numbers less than $100$ that share the same prime factors.

Potential Misunderstandings

• Students may stop factor trees before reaching prime numbers
• Students may include composite numbers in final factorisation
• Students may forget repeated prime factors
• Students may incorrectly convert repeated factors into powers
• Students may think different factor trees give different prime factorisations
• Students may confuse factors with prime factors

Next: 016. Squares, Roots and Perfect Squares