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Week 1Pillar 1 · NumbersModule 1.1

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.

72236492, 235HCF = shared core
Prime factor tree with intersecting Venn factor sets
  • •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 · Homework practice (10 questions)

  1. Bronze1

    List all factors of 60.

    Working Space & Algebraic Steps:
    Final Answer:
  2. Bronze2

    Write 126 as a product of primes in index form.

    Working Space & Algebraic Steps:
    Final Answer:
  3. Bronze3

    State whether 87 is prime, with justification.

    Working Space & Algebraic Steps:
    Final Answer:
  4. Silver4

    Find HCF(140, 196) using prime factorisation.

    Working Space & Algebraic Steps:
    Final Answer:
  5. Silver5

    Find LCM(12, 20, 45).

    Working Space & Algebraic Steps:
    Final Answer:
  6. Silver6

    Two buses leave a station every 18 minutes and every 30 minutes. They left together at 07:00, when will they next leave together?

    Working Space & Algebraic Steps:
    Final Answer:
  7. Silver7

    The HCF of two numbers is 6 and their LCM is 180. One number is 36. Find the other.

    Working Space & Algebraic Steps:
    Final Answer:
  8. Gold8

    A rectangular floor measures 360 cm by 600 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:
  9. Gold9

    N = 2² × 3³ × 5. How many factors does N have, and how many are even?

    Working Space & Algebraic Steps:
    Final Answer:
  10. Gold10

    Find the smallest positive integer k such that 1 350 × 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.