ACARA v9 CONTENT DESCRIPTION “identify and describe the properties of prime, composite, square and triangular numbers and use these properties to solve problems and simplify calculations”
Builds on: Decimals to Thousandths (AC9M5N01). Multiplication and division facts from earlier years are the tools for finding factors and multiples — here they reveal the hidden structure inside whole numbers.
Every number has factors
A factor of a number is a whole number that divides it exactly, with nothing left over. The number 12 has factors 1, 2, 3, 4, 6 and 12, because each one fits into 12 a whole number of times. A neat way to see factors is as rectangles: 12 squares can be arranged as 1 by 12, 2 by 6 or 3 by 4, and each rectangle that fills completely shows a factor pair. Every number can be built this way, and reading off its factors is the first step to understanding how numbers are put together.
Factors build rectangles
A factor pair of a number is the sides of a rectangle holding exactly that many squares.
12 has factors 1, 2, 3, 4, 6, 12 — each rectangle that fills exactly shows one factor pair.
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Multiples count in steps
Where factors look inside a number, multiples look outward. The multiples of a number are what you reach by counting up in equal steps: the multiples of 3 are 3, 6, 9, 12 and on, and the multiples of 5 are 5, 10, 15, 20. Every number has endlessly many multiples, one for each step you take. Factors and multiples are two sides of the same relationship: 3 is a factor of 12, and 12 is a multiple of 3. Holding both ideas together is what makes the rest of this unit work.
Multiples are counting steps
The multiples of a number are what you land on counting up in equal jumps.
The multiples of 3 are 3, 6, 9, 12, ... — the numbers you reach counting in steps of 3.
Primes and composites
Some numbers can be made into only one rectangle, a single row. A prime number has exactly two factors, 1 and itself, so 7, 13 and 23 are prime: they cannot be split into smaller equal groups. A composite number has more than two factors, like 12 or 15, so it can be arranged as several different rectangles. The number 1 is neither, having just one factor. Sorting numbers into prime and composite reveals which are the indivisible building blocks and which are built from them.
Prime or composite
Two factors only means prime. More than two means composite.
Is 13 prime or composite? A prime has exactly two factors; a composite has more.
Factors two numbers share
When two numbers are compared, the factors they have in common matter. Both 12 and 18 have 1, 2, 3 and 6 among their factors, so these are their common factors, and the largest of them, 6, is the highest common factor. Finding the highest common factor is what lets a fraction be simplified to its lowest terms, or a quantity be shared into the largest equal groups. It is one of the most useful things factors are good for, turning two separate lists into a single shared answer.
Factors two numbers share
List each number's factors; the ones in both are common factors.
12 and 18 share factors 1, 2, 3, 6 — the largest, 6, is their highest common factor.
Where multiples meet
Multiples lead to a matching idea. Counting in fours gives 4, 8, 12, 16; counting in sixes gives 6, 12, 18; the first number that appears in both is 12, the lowest common multiple. The lowest common multiple is what you need to add fractions with different denominators, or to work out when two repeating events line up again. Just as the highest common factor is the largest factor two numbers share, the lowest common multiple is the smallest multiple they share, and the two ideas work as a pair.
Where multiples first meet
The lowest common multiple is the first number that appears in both counting sequences.
Counting in 4s and in 6s, both first meet at 12 — the lowest common multiple.
Breaking down into primes
Every composite number can be broken down into primes, and in only one way. Splitting 12 into 2 times 6, then 6 into 2 times 3, leaves 2, 2 and 3, all prime: this is the prime factorisation of 12. A factor tree is a tidy way to carry out the splitting, branching each number into factors until only primes remain at the tips. Because this breakdown is unique to each number, prime factorisation is a kind of fingerprint, and it underlies how highest common factors and lowest common multiples can be found quickly.
Breaking down into primes
Split a number into factors until only primes remain. That is its prime factorisation.
12 breaks down into the primes 2, 2, 3 — every whole number is a product of primes.
Why this structure matters
Factors, multiples and primes are the grammar of whole numbers. Knowing them lets you simplify fractions, find common denominators, solve sharing and grouping problems, and recognise the patterns that square and triangular numbers follow. These ideas reach forward into every later topic that uses number: ratio, algebra and beyond all rest on being able to see a number not just as a quantity, but as something with an inner structure of factors and primes waiting to be used.
Quick self-check
1. Which list gives all the factors of 12?
2. Which number is a multiple of 7?
3. Which of these is a prime number?
4. What is the highest common factor of 12 and 18?