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Sec 1 Factors, Multiples, HCF and LCM | Prime Factorisation and Problem Solving

Secondary 1 Factors, Multiples, HCF and LCM is not just a revision of Primary number facts. At Secondary level, students need to see the structure connecting divisibility, prime factorisation, common factors and common multiples. That structure becomes useful later in algebraic factorisation, fractions and number theory.

Students searching for HCF LCM Sec 1, factors and multiples, prime factorisation or Secondary 1 Mathematics often memorise separate methods for HCF and LCM. The more durable system is factorise → compare prime powers → select common or required powers → rebuild the answer.

This page is the canonical Factors/Multiples/HCF/LCM owner under the Secondary Mathematics Topic Library.

Quick answer: the four ideas

  • Factor: divides a number exactly.
  • Multiple: result of multiplying a number by an integer.
  • HCF: greatest factor common to all given numbers.
  • LCM: smallest positive multiple common to all given numbers.

Prime factorisation is the common language

Every whole number greater than 1 can be written as a product of primes.

Example: 84 = 2² × 3 × 7.

Prime factorisation makes HCF and LCM systematic instead of list-based.

Factor trees

A factor tree breaks a composite number into factors until only primes remain.

Different factor trees for the same number end with the same prime factors, apart from order. This is a useful check.

Worked example 1: HCF

Find HCF of 72 and 90.

72 = 2³ × 3².

90 = 2 × 3² × 5.

Common prime powers at the smallest available exponents: 2 × 3² = 18.

HCF = 18.

Worked example 2: LCM

Find LCM of 72 and 90.

Use every required prime at the highest exponent: 2³ × 3² × 5 = 360.

LCM = 360.

The prime-power rule

  • HCF: take only primes common to all numbers, using the smallest exponent.
  • LCM: take every prime needed, using the largest exponent.

This rule becomes much easier when students understand that HCF must fit inside every number, while LCM must contain enough factors to be divisible by every number.

Worked example 3: three numbers

Find HCF and LCM of 24, 36 and 60.

24 = 2³ × 3.

36 = 2² × 3².

60 = 2² × 3 × 5.

HCF = 2² × 3 = 12.

LCM = 2³ × 3² × 5 = 360.

HCF problem language

HCF often appears when a quantity must be split into the largest equal groups with no remainder.

Example: 48 red beads and 60 blue beads are packed into identical sets using all beads. What is the greatest number of sets? HCF(48,60)=12.

LCM problem language

LCM often appears when repeating cycles need to coincide again.

Example: One bell rings every 12 minutes and another every 18 minutes. If they ring together now, when will they next ring together? LCM(12,18)=36 minutes.

Do not rely on keywords alone

“Greatest” does not always mean HCF and “smallest” does not always mean LCM. Understand the structure: are we partitioning into equal groups, or synchronising repeated multiples?

Divisibility checks

  • divisible by 2: last digit even
  • divisible by 3: digit sum divisible by 3
  • divisible by 5: ends in 0 or 5
  • divisible by 9: digit sum divisible by 9
  • divisible by 10: ends in 0

Divisibility rules speed factorisation and provide quick checks.

Common HCF/LCM errors

  • takes largest exponents for HCF
  • takes smallest exponents for LCM
  • forgets a prime that appears in only one number when finding LCM
  • includes a non-common prime in HCF
  • confuses equal grouping with repeated-cycle questions
  • prime factorisation arithmetic error

The relationship for two numbers

For two positive integers a and b:

HCF(a,b) × LCM(a,b) = a × b.

This can be used as a check or to find one quantity if the others are known.

Worked example 4: use the relationship

Two numbers are 18 and 30.

HCF = 6. LCM = 90.

Check: 6 × 90 = 540 and 18 × 30 = 540.

How this prepares for algebra

Factoring numbers prepares students to factor algebraic expressions. For example, HCF of 12 and 18 is 6; similarly, the common factor of 12x and 18x² begins with 6x.

Number structure becomes algebraic structure.

Practice

Practice 1

Question: HCF of 18 and 24

Answer: 6

Practice 2

Question: LCM of 18 and 24

Answer: 72

Practice 3

Question: HCF of 42, 56, 70

Answer: 14

Practice 4

Question: LCM of 6, 8, 15

Answer: 120

Practice 5

Question: Bells every 8 and 12 min

Answer: together every 24 min

Practice 6

Question: Largest equal sets from 36 and 48 items

Answer: 12 sets

Frequently asked questions

What is the difference between a factor and a multiple?

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

How do I remember HCF exponents?

HCF can use only what every number has, so take the smallest common exponent.

How do I remember LCM exponents?

LCM must contain enough of every prime to cover all numbers, so take the largest exponent.

When should I use HCF in word problems?

When dividing quantities into the greatest possible equal groups with no remainder.

When should I use LCM?

When repeated cycles or multiples must meet again.

Where does this sit in Atlas?

This is the canonical Sec 1 Factors/Multiples/HCF/LCM owner under Secondary Mathematics Topic Library.

The final HCF/LCM rule

Prime factorisation is the map. HCF takes the shared structure every number contains; LCM builds the smallest structure large enough to contain them all.

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