2.6 Factors, Multiples, Primes & Divisibility

Factors, Multiples, Primes & Divisibility

The official familiarisation booklet says the maths section covers "topics taught in most schools up to the start of Year 6", with a small number of harder items that ask you to apply familiar skills to new problems. Number properties are exactly that kind of topic: the facts are ordinary Year 5 content, but the questions bury them inside a puzzle. A child who knows the facts cold answers in seconds; a child who has to work them out loses a minute they do not have.

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The Core Vocabulary

Two facts to nail down because they are asked directly:

• 1 is not prime. It has only one factor, and the definition requires exactly two.
• 2 is the only even prime. Every other even number has 2 as a third factor.

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Finding Factors Without Missing Any

Factors come in pairs, so list them in pairs working outwards from 1 and stop when the pair meets:

For 36: 1 × 36, 2 × 18, 3 × 12, 4 × 9, 6 × 6 → stop, because 6 × 6 has met itself. Factors: 1, 2, 3, 4, 6, 9, 12, 18, 36 — nine of them.

A square number always has an odd number of factors, because one pair is a repeat. That is a favourite exam question: "which of these has an odd number of factors?" means "which of these is a square number?"

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Divisibility Rules

These convert a division problem into a glance.

Worked use: "Which of these is not a multiple of 3? A 471 B 582 C 634 D 726 E 915." Digit sums: 12, 15, 13, 15, 15. Answer C, found in about five seconds with no division at all.

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Prime Factorisation

Every whole number above 1 breaks into primes in exactly one way. Use a factor tree and finish in index form.

180 → 18 × 10 → (2 × 9) × (2 × 5) → 2 × 3 × 3 × 2 × 5 180 = 2² × 3² × 5

Prime factorisation also counts factors: add 1 to each index and multiply. For 180: (2+1)(2+1)(1+1) = 3 × 3 × 2 = 18 factors. That beats listing them when the number is large.

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HCF and LCM the Fast Way

Write both numbers in index form, then:

• HCF (highest common factor): take each shared prime to its lowest power.
• LCM (lowest common multiple): take each prime that appears in either number to its highest power.

48 = 2⁴ × 3 and 180 = 2² × 3² × 5 HCF = 2² × 3 = 12 LCM = 2⁴ × 3² × 5 = 720

The check that catches errors: HCF × LCM = the product of the two numbers. Here 12 × 720 = 8,640, and 48 × 180 = 8,640. ✓

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How These Appear as Word Problems

Number-property content almost never arrives labelled. It arrives like this:

• "Two lighthouses flash every 12 seconds and every 18 seconds. They flash together now. After how many seconds do they next flash together?" → LCM of 12 and 18 = 36 seconds.
• "A baker has 42 scones and 56 muffins and wants identical boxes with nothing left over. What is the largest number of boxes?" → HCF of 42 and 56 = 14 boxes.
• "I am a two-digit number, one less than a square number, and a multiple of 9. What am I?" → one less than a square: 15, 24, 35, 48, 63, 80, 99. Apply the digit-sum test for 9: only 63 (6+3 = 9) and 99 (9+9 = 18) qualify, so the answer will be whichever the options offer. Generate the smaller list first — there are only seven two-digit numbers one less than a square, but eleven multiples of 9.

The signal words are consistent. "Together again", "at the same time", "next occurs" means LCM. "Largest group", "identical shares", "nothing left over" means HCF.