Express natural numbers as products of their factors, recognise multiples and determine if one number is divisible by another

Students distinguish factors from multiples, generate factor pairs systematically and use multiplication facts and divisibility reasoning to decide whether division gives a whole-number result.

A factor divides a number exactly; a multiple is produced by multiplying the number by a whole number.

Stop when factor pairs repeat in reverse. The factors of 36 are 1, 2, 3, 4, 6, 9, 12, 18 and 36.

A divisibility test is a reasoned shortcut. Confirm unfamiliar cases with multiplication or division rather than relying on a guessed pattern.

Learning routine: Represent → Reason → Calculate → Interpret → Verify

Success looks like

• Represent the concept accurately
• Explain the place-value or number relationship
• Select an efficient strategy
• Apply it to a new context
• Verify and justify the result

Content description: express natural numbers as products of their factors, recognise multiples and determine if one number is divisible by another.

• E1: using a certain number of blocks to form different rectangles and using these to list all possible factors for that number; for example, 12 blocks can form the following rectangles: 1 \times 12, 2 \times 6, and 3 \times 4
• E2: researching divisibility tests and explaining each rule using materials; for example, using base-10 blocks to test if numbers are divisible by 2, 5 and 10
• E3: using divisibility tests to determine if larger numbers are multiples of one-digit numbers; for example, testing if 89 472 is divisible by 3 using 8 + 9 + 4 + 7 + 2=30 as 30 is divisible by 3 then 89 472 is a multiple of 3
• E4: demonstrating and reasoning that all multiples can be formed by combining or regrouping; for example, multiples of 7 can be formed by combining a multiple of 2 with the corresponding multiple of 5; 3 \times 7 = 3 \times 2 + 3 \times 5, and 4 \times 7 = 4 \times 2 + 4 \times 5

Core idea: A factor divides a number exactly; a multiple is produced by multiplying the number by a whole number.

Remember

• Represent the concept accurately
• Explain the place-value or number relationship
• Select an efficient strategy
• Apply it to a new context
• Verify and justify the result

Important questions

• List factors of 24. Explain using the model or evidence above.
• Give five multiples of 7. Explain using the model or evidence above.
• Is 126 divisible by 9? Check. Explain using the model or evidence above.
• Find a common multiple of 6 and 8. Explain using the model or evidence above.
• Explain factor versus multiple. Explain using the model or evidence above.

• Factors and multiples reversed — Factors fit into a number; multiples extend outward from it.
• Only small factor pairs listed — Continue systematically until the pair order repeats.
• Divisible means close to a multiple — The quotient must be a whole number with no remainder.
• One is forgotten — 1 and the number itself are factors of every positive natural number.

Learn from the Topic Guide and fixed Teacher Slides, complete the Practice Sheet, use Practice for supported feedback, then take the Test when ready.

Students distinguish factors from multiples, generate factor pairs systematically and use multiplication facts and divisibility reasoning to decide whether division gives a whole-number result.

A factor divides a number exactly; a multiple is produced by multiplying the number by a whole number.

Stop when factor pairs repeat in reverse. The factors of 36 are 1, 2, 3, 4, 6, 9, 12, 18 and 36.

A divisibility test is a reasoned shortcut. Confirm unfamiliar cases with multiplication or division rather than relying on a guessed pattern.

Learning routine: Represent → Reason → Calculate → Interpret → Verify

Success looks like

• Represent the concept accurately
• Explain the place-value or number relationship
• Select an efficient strategy
• Apply it to a new context
• Verify and justify the result

Australian Curriculum Version 9.0

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