013. Prime Numbers and Prime Factorisation

Learning Intentions

• To know what prime numbers and composite numbers are
• determine whether a number is prime by considering its factors
• find the prime factors of a given number

Pre-requisite Summary

• Understanding factors and multiples
• Knowledge that $1$ and the number itself are always factors
• Ability to test divisibility by small numbers ($2,3,4,5,6,8,9,10$)
• Understanding multiplication as forming composite numbers
• Familiarity with division and factor pairs

Worked Examples

Worked Example 1

Classify numbers as prime or composite:

a) $7$
b) $12$
c) $19$

Worked Example 2

Determine if the number is prime by considering factors:

a) $21$
b) $29$
c) $31$

Worked Example 3

Find prime factors:

a) $18$
b) $24$

Worked Example 4

Find prime factors:

a) $45$
b) $84$

Problems

Problem 1

Classify as prime or composite:

a) $11$
b) $15$
c) $23$

Problem 2

Determine if prime:

a) $27$
b) $37$
c) $41$

Problem 3

Find prime factors:

a) $20$
b) $36$

Problem 4

Find prime factors:

a) $54$
b) $72$

Exercises

Understanding and Fluency

1. Classify as prime or composite:
   a) $2$
   b) $9$
   c) $17$

2. Classify as prime or composite:
   a) $13$
   b) $21$
   c) $25$

3. Determine if prime by considering factors:
   a) $33$
   b) $47$
   c) $51$

4. Determine if prime by considering factors:
   a) $39$
   b) $43$
   c) $57$

5. Find prime factors:
   a) $16$
   b) $28$
   c) $40$

6. Find prime factors:
   a) $50$
   b) $63$
   c) $90$

Reasoning

7. Explain why $1$ is not a prime number.

8. Why must every composite number have at least one prime factor?

9. A student says $15$ is prime because it is odd. Explain the mistake.

10. Explain why $2$ is the only even prime number.

Problem-solving

11. Find two different composite numbers that have the same prime factors.

12. Find a number whose prime factors are $2$ and $3$.

13. Find all prime numbers between $20$ and $30$.

14. A number has prime factorisation $2\times 2\times 3$. What is the number?

15. Find a number less than $50$ with three prime factors.

Potential Misunderstandings

• Students may think $1$ is a prime number
• Students may think all odd numbers are prime
• Students may stop checking factors too early
• Students may include composite numbers when listing prime factors
• Students may forget to continue factorising until only primes remain
• Students may confuse prime factorisation with listing all factors

Next: 014. Powers and Index Notation