2F Prime decomposition

Learning intentions for this section:

• To understand that composite numbers can be broken down into a product of prime factors

• To be able to find the prime factors of a number (including repeated factors)

• To be able to express a prime decomposition using powers of prime numbers

Past, present and future learning:

• The concepts in this section may be new to students

• These concepts are assumed knowledge for future learning beyond Stage 4

All composite numbers can be broken down (i.e. decomposed) into a unique set of prime factors. A common way of performing the decomposition into prime factors is using a factor tree. Starting with the given number, ‘branches’ come down in pairs, representing a pair of factors that multiply to give the number above it. This process continues until prime factors are reached.

Lesson starter: Composition of numbers from prime factors

‘Compose’ composite numbers by multiplying the following sets of prime factors. The first one has been done for you.

a 2 × 3 × 5 = 30 b 2 × 3 × 7 × 3 × 2 c 32 × 23 e 13 × 17 × 2 f 22 × 52 × 72 g 25 × 34 × 7

Key Ideas

d 5 × 11 × 22 h 11 × 13 × 17

The reverse of this process is called decomposition, where a number like 30 is broken down into the prime factors 2 × 3 × 5.

■ Every composite number can be expressed as a product of its prime factors

■ A factor tree can be used to show the prime factors of a composite number.

■ Each ‘branch’ of a factor tree eventually terminates in a prime factor.

■ Powers are often used to efficiently represent composite numbers in prime factor form. For example: