2.3 Factors, Multiples & Prime Numbers

From Paper C onwards ICAS asks about factors and multiples directly, and Papers E–F add index notation and prime factorisation. These questions are quick marks if your times tables are automatic — and slow, error-prone guesses if they are not. The ideas also power later topics: simplifying fractions uses the HCF, and adding fractions uses the LCM.

Factors vs Multiples

A factor of a number divides it exactly, with no remainder. A multiple is what you get by multiplying the number by 1, 2, 3, … — its times-table answers.

Two rules to memorise: factors are never larger than the number itself, and every number is both a factor and a multiple of itself. ICAS traps include answers like "24 is a factor of 12" — 24 is a multiple of 12, never a factor.

Factor Pairs

List factors in pairs that multiply to the number, working up from 1. For 36: 1 × 36, 2 × 18, 3 × 12, 4 × 9, 6 × 6. Stop when a pair repeats — the factors of 36 are 1, 2, 3, 4, 6, 9, 12, 18, 36. Notice 6 × 6: square numbers have an odd number of factors because one pair collapses into a single repeated factor.

Common Factors, HCF, Common Multiples, LCM

Common factors appear in both numbers' factor lists. The highest common factor (HCF) — sometimes called the greatest common divisor — is the largest of them.

Worked example: HCF of 24 and 36.

• Factors of 24: 1, 2, 3, 4, 6, 8, 12, 24.
• Factors of 36: 1, 2, 3, 4, 6, 9, 12, 18, 36.
• Common: 1, 2, 3, 4, 6, 12. The HCF is 12.

The lowest common multiple (LCM) is the smallest number in both times tables.

Worked example: LCM of 6 and 8.

• Multiples of 6: 6, 12, 18, 24, 30, …
• Multiples of 8: 8, 16, 24, 32, …
• The LCM is 24.

A useful check for two numbers: HCF × LCM = product of the numbers. For 24 and 36: 12 × 72 = 864 = 24 × 36 ✓. ICAS word problems disguise these: "hot dogs in packs of 6, buns in packs of 8 — smallest equal purchase?" is an LCM question (24 of each).

Prime and Composite Numbers

A prime number has exactly two factors: 1 and itself. A composite number has more than two. Memorise the primes to 100 — there are 25 of them:

2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97.

Key facts ICAS tests directly:

• 2 is the only even prime — every other even number has 2 as a factor.
• 1 is not prime — it has only one factor, and primes need exactly two.
• 9, 15 and 21 are not prime despite looking "odd and awkward": 9 = 3 × 3, 15 = 3 × 5, 21 = 3 × 7.

Square, Triangular and Cube Numbers

Paper E adds special families of numbers alongside primes and composites, and ICAS tests them by asking you to recognise, continue or apply them rather than to define them.

Square numbers come from multiplying a whole number by itself: 1, 4, 9, 16, 25, 36, 49, 64, 81, 100, 121, 144. Learn them at least to 15² = 225 — they recur in area work, in Pythagoras (Section 11.3) and in index laws (Section 4.3). Two properties ICAS likes to test:

• A square number has an odd number of factors, because one factor pair collapses into a repeat. For 36: 1 × 36, 2 × 18, 3 × 12, 4 × 9 and 6 × 6, giving the nine factors 1, 2, 3, 4, 6, 9, 12, 18, 36.
• Consecutive squares differ by successive odd numbers: 1, 4, 9, 16 differ by 3, 5 and 7. So the square after 144 is 144 + 25 = 169 = 13².

Triangular numbers count the dots in a growing triangular arrangement: 1, 3, 6, 10, 15, 21, 28, 36, 45, 55. Each term adds the next whole number, so the gaps grow 2, 3, 4, 5, … The nth triangular number is

T(n) = n(n + 1) ÷ 2

so T(10) = 10 × 11 ÷ 2 = 55. This formula also answers the classic handshake question: if everyone in a group of 11 people shakes hands once with everyone else, the number of handshakes is T(10) = 55. ICAS disguises triangular numbers as dot patterns, stacked cans or seating arrangements far more often than it names them.

Worked example. Which numbers below 50 are both square and triangular? The squares are 1, 4, 9, 16, 25, 36, 49 and the triangular numbers are 1, 3, 6, 10, 15, 21, 28, 36, 45. The overlap is 1 and 36. For any 'which number is both …' question, list the shorter family first and test each member against the other.

Cube numbers complete the set: 1, 8, 27, 64, 125, formed by multiplying a number by itself three times.

Divisibility Tests

These let you find factors without dividing:

The digit-sum tests are the most powerful. To check whether 87 is prime: 8 + 7 = 15, which is divisible by 3, so 87 = 3 × 29 — composite, no division needed.

Prime Factorisation with Factor Trees

Every whole number greater than 1 can be written as a product of primes in exactly one way — this is its prime factorisation. Build a factor tree: split the number into any factor pair, then keep splitting every composite branch until only primes remain.

Worked example: 360.

• 360 = 10 × 36
• 10 = 2 × 5; 36 = 6 × 6
• Each 6 = 2 × 3
• Collecting the prime leaves: 360 = 2 × 2 × 2 × 3 × 3 × 5

Any first split works — 360 = 4 × 90 leads to the same leaves. That uniqueness is guaranteed, so a factor tree is self-checking: multiply your primes back together to confirm.

Index Notation (Papers E–F)

Papers E and F expect the answer written with indices (powers): repeated primes are gathered into a single base with an exponent counting how many times it appears.

• 360 = 2 × 2 × 2 × 3 × 3 × 5 = 2³ × 3² × 5.
• Read 2³ as "two cubed" = 8; 3² = "three squared" = 9. Then 8 × 9 × 5 = 360 ✓.

Index notation also speeds up HCF and LCM for larger numbers: take the lowest power of each shared prime for the HCF, and the highest power of every prime for the LCM. With 24 = 2³ × 3 and 36 = 2² × 3²: HCF = 2² × 3 = 12 and LCM = 2³ × 3² = 72 — matching our earlier list method.

How ICAS Phrases These Questions

• "Which of the following is a factor of both 24 and 36?" — check each option divides both.
• "Which number between 40 and 50 is prime?" — test 41 (prime), 42, 43 (prime), 44–49 with the divisibility tests.
• "A number is written as 2² × 3 × 5. What is the number?" — 4 × 3 × 5 = 60.
• "Which of these numbers is divisible by both 3 and 5?" — it must be divisible by 15: last digit 0 or 5 and digit sum divisible by 3.