Factors, Multiples & Primes
Target mastery benchmark
The Fundamental Theorem of Arithmetic, canonical factorisation, HCF and LCM
1 · Concept introduction
Every integer greater than 1 is either prime or can be written as a product of primes in exactly one way. This is the Fundamental Theorem of Arithmetic, and it is the reason prime factorisation is a reliable structural fingerprint of a number.
A prime factor tree splits a number repeatedly until every branch terminates in a prime. Collecting the terminal primes and writing repeated primes as powers gives the canonical form, e.g. 72 = 2³ × 3².
The Highest Common Factor (HCF) is built from the primes the numbers share, each taken to its lowest power. The Lowest Common Multiple (LCM) takes every prime that appears in either number, each to its highest power.
Because HCF collects the shared core and LCM collects the full union, HCF(a, b) × LCM(a, b) = a × b for any two positive integers.
- •N = p₁^a × p₂^b × p₃^c … (canonical prime form)
- •HCF: shared primes to the lowest power
- •LCM: all primes to the highest power
- •HCF(a, b) × LCM(a, b) = a × b
2 · Core theory & worked examples
Worked Example 1 — Canonical form of 360
Express 360 as a product of primes in index form.
- 1Divide by the smallest prime repeatedly: 360 ÷ 2 = 180, 180 ÷ 2 = 90, 90 ÷ 2 = 45.
- 245 is odd, so move to the next prime: 45 ÷ 3 = 15, 15 ÷ 3 = 5.
- 35 is prime, so the tree terminates.
- 4Collect terminal primes: 2 × 2 × 2 × 3 × 3 × 5.
- 5Write in index form: 360 = 2³ × 3² × 5.
Core Heuristic: always divide by the smallest prime that works, and only move up when the current prime no longer divides. This forces the canonical order automatically.
Worked Example 2 — HCF and LCM of 84 and 120
Find HCF(84, 120) and LCM(84, 120).
- 1Factorise: 84 = 2² × 3 × 7 and 120 = 2³ × 3 × 5.
- 2Shared primes are 2 and 3. Lowest powers: 2² and 3¹.
- 3HCF = 2² × 3 = 12.
- 4All primes present: 2, 3, 5, 7. Highest powers: 2³, 3¹, 5¹, 7¹.
- 5LCM = 8 × 3 × 5 × 7 = 840.
- 6Check: 12 × 840 = 10 080 = 84 × 120. ✓
Core Heuristic: lay both factorisations in a column aligned by prime. HCF reads the minimum of each row, LCM reads the maximum.
3 · In-class guided practice (10 questions)
- Bronze1
List all factors of 48.
Working Space & Algebraic Steps:Final Answer: - Bronze2
Write 90 as a product of primes in index form.
Working Space & Algebraic Steps:Final Answer: - Bronze3
State whether 91 is prime, with justification.
Working Space & Algebraic Steps:Final Answer: - Silver4
Find HCF(126, 210) using prime factorisation.
Working Space & Algebraic Steps:Final Answer: - Silver5
Find LCM(18, 24, 30).
Working Space & Algebraic Steps:Final Answer: - Silver6
Two lighthouses flash every 24 s and every 36 s. They flashed together at 21:00, when will they next flash together?
Working Space & Algebraic Steps:Final Answer: - Silver7
The HCF of two numbers is 8 and their LCM is 96. One number is 24. Find the other.
Working Space & Algebraic Steps:Final Answer: - Gold8
A rectangular floor measures 315 cm by 450 cm. It is to be covered completely with identical square tiles, laid in rows and columns with no tiles cut. Find the side length of the largest possible square tile, and the number of tiles needed.
Working Space & Algebraic Steps:Final Answer: - Gold9
N = 2³ × 3² × 5. How many factors does N have, and how many are even?
Working Space & Algebraic Steps:Final Answer: - Gold10
Find the smallest positive integer k such that 1 176 × k is a perfect cube.
Working Space & Algebraic Steps:Final Answer:
Self-audit checkpoint
- I can build a prime factor tree and write the result in index form.
- I can extract HCF and LCM from aligned factorisations without guessing.
- I can decide when a word problem needs HCF (splitting/grouping) versus LCM (repeating cycles).
- I can use HCF × LCM = a × b as a checking tool.