Learning Objectives
  • Classify natural numbers, integers, prime numbers, square numbers and cube numbers.
  • Write a positive integer as a product of prime factors using index notation.
  • Find the highest common factor and lowest common multiple of two or more integers.
  • Select efficient methods for divisibility, factor and multiple problems.
Key Terms
Factor
A whole number that divides another whole number exactly.
Multiple
A number obtained by multiplying a given number by an integer.
Prime number
A positive integer greater than 1 with exactly two positive factors: 1 and itself.
Prime factorisation
Writing a number as a product of prime numbers.
HCF
The greatest positive integer that is a factor of every number under consideration.
LCM
The smallest positive integer that is a multiple of every number under consideration.
Types Of Number And Divisibility

The natural numbers are the counting numbers 1,2,3,\ldots. Integers include positive whole numbers, zero and negative whole numbers. A factor divides a number exactly, while a multiple appears in that number’s multiplication sequence. For example, the factors of 18 are 1,2,3,6,9,18, and some multiples of 18 are 18,36,54,72,\ldots.

A prime number has exactly two positive factors. The number 1 is not prime because it has only one positive factor. The number 2 is the only even prime number. Composite numbers have more than two positive factors and can be broken into smaller factors.

Divisibility test Condition
2 The final digit is even.
3 The sum of the digits is divisible by 3.
4 The number formed by the last two digits is divisible by 4.
5 The final digit is 0 or 5.
6 The number is divisible by both 2 and 3.
8 The number formed by the last three digits is divisible by 8.
9 The sum of the digits is divisible by 9.
10 The final digit is 0.
Prime Factorisation

Every positive integer greater than 1 can be written as a product of primes. A factor tree is a useful method, but different correct factor trees always lead to the same prime factors. The final answer should normally be written using indices.

360=2^3\times 3^2\times 5
Worked Example: Prime Factorisation

Question: Express 756 as a product of prime factors.

  1. Divide by 2: 756=2\times378.
  2. Divide by 2 again: 378=2\times189.
  3. Then 189=3\times63=3\times3\times21=3\times3\times3\times7.
  4. Collect equal factors: 756=2^2\times3^3\times7.

Answer: 2^2\times3^3\times7

Highest Common Factor

The HCF contains only prime factors common to every number, using the smallest power that occurs. It is useful when making the largest possible equal groups, cutting equal lengths without waste or simplifying a ratio.

Worked Example: HCF

Question: Find the HCF of 84 and 126.

  1. Write 84=2^2\times3\times7.
  2. Write 126=2\times3^2\times7.
  3. Choose the common primes with the smaller powers: 2^1\times3^1\times7^1.
  4. Multiply: 2\times3\times7=42.

Answer: The HCF is 42.

Lowest Common Multiple

The LCM contains every prime factor required by any number, using the greatest power that occurs. It is useful when finding when repeating events occur together or when finding a common denominator.

Worked Example: LCM

Question: Find the LCM of 72 and 90.

  1. Write 72=2^3\times3^2.
  2. Write 90=2\times3^2\times5.
  3. Choose every prime with the greater power: 2^3\times3^2\times5.
  4. Multiply: 8\times9\times5=360.

Answer: The LCM is 360.

Problem-Solving With HCF And LCM

Look for language that reveals the method. “Largest equal size”, “greatest possible length” and “without remainder” usually indicate HCF. “First time together”, “smallest number divisible by” and “common denominator” usually indicate LCM.

Worked Example: Repeating Events

Question: A bus leaves every 18 minutes and another every 24 minutes. They leave together at 08:00. When will they next leave together?

  1. Find \operatorname{LCM}(18,24).
  2. Use 18=2\times3^2 and 24=2^3\times3.
  3. The LCM is 2^3\times3^2=72 minutes.
  4. Add 72 minutes to 08:00.

Answer: They next leave together at 09:12.

8th Edition Chapter Map
  • Prime numbers and prime factorisation
  • Squares, square roots, cubes and cube roots
  • Highest common factor and lowest common multiple
Squares, Cubes And Their Roots

A square number is the result of multiplying an integer by itself. The first positive square numbers are 1,4,9,16,25,36,49,64,81,100. A cube number is the result of multiplying an integer by itself three times: 1,8,27,64,125,216,343,512,729,1000. Recognising these values quickly is useful in factorisation, estimation and later algebra.

The principal square root of a positive number is the non-negative number that squares to give it. Thus \sqrt{144}=12. An equation such as x^2=144 has two solutions, x=12 and x=-12, but the symbol \sqrt{144} means the positive root only. A cube root keeps the sign of the original number, so \sqrt[3]{-125}=-5.

Worked Example: Finding Roots By Prime Factors

Question: Find \sqrt{1764} without using trial and error.

  1. Prime-factorise: 1764=2^2\times3^2\times7^2.
  2. A square root takes one factor from each pair.
  3. \sqrt{1764}=2\times3\times7=42.

Answer: 42.

Efficient HCF And LCM Reasoning

Prime-factor methods are systematic, especially for three numbers or large values. For the HCF, keep only factors present in every number and use the smallest exponent. For the LCM, include every factor needed and use the largest exponent. The product rule \operatorname{HCF}(a,b)\times\operatorname{LCM}(a,b)=a\times b is useful as a check for two positive integers.

Worked Example: Three Numbers

Question: Find the HCF and LCM of 72,108 and 180.

  1. 72=2^3\times3^2
  2. 108=2^2\times3^3
  3. 180=2^2\times3^2\times5
  4. HCF: smallest common powers 2^2\times3^2=36.
  5. LCM: largest powers 2^3\times3^3\times5=1080.
Choosing The Correct Tool In Context
Clue in the question Likely method Reason
largest equal groups, longest equal pieces HCF The group size must divide every quantity.
first time together, smallest common total LCM The answer must be a multiple of every cycle or quantity.
is a large number prime? Test divisibility up to its square root A composite number has a factor not exceeding its square root.
exact square or cube root Recognise known values or pair/group prime factors Each pair gives one square-root factor; each group of three gives one cube-root factor.
Worked Example: Packing Without Waste

Red beads come in a box of 84 and blue beads in a box of 126. Identical activity packs must contain the same number of red beads and the same number of blue beads, with no beads left. The greatest possible number of packs is \operatorname{HCF}(84,126)=42. Each pack contains 84\div42=2 red beads and 126\div42=3 blue beads.

Extended Practice

A. Find \sqrt[3]{2744}.

2744=14^3, so the answer is 14.

B. The HCF of two numbers is 12, their LCM is 420 and one number is 60. Find the other number.

Using the product rule, the other number is (12\times420)\div60=84.

C. Bells ring every 24, 36 and 54 minutes. They ring together at 09:00. When do they next ring together?

\operatorname{LCM}(24,36,54)=216 minutes, which is 3 hours 36 minutes. The time is 12:36.

Examination Guidance
  • Show prime factorisations clearly when the question asks for HCF or LCM by prime factors.
  • Do not list 1 as a prime number.
  • Check whether the context asks for a greatest possible size (HCF) or a first common repetition (LCM).
  • When a final answer represents people, boxes or groups, ensure it is a whole number and include the correct unit or description.
Common Mistakes
  • Confusing factors with multiples.
  • Using the greatest power for HCF instead of the smallest common power.
  • Stopping a factor tree before every branch ends in a prime number.
  • Giving a prime factorisation such as 4\times3\times5 instead of fully factorising 4.
Knowledge Check

1. Write 540 as a product of prime factors.

Answer

540=2^2\times3^3\times5.

2. Find the HCF of 96 and 144.

Answer

48.

3. Find the LCM of 28 and 42.

Answer

84.

4. Explain why 1 is not prime.

Answer

It has only one positive factor, not exactly two.

5. Three lights flash every 12, 18 and 30 seconds. After how many seconds do they next flash together?

Answer

\operatorname{LCM}(12,18,30)=180 seconds.

Mixed Review With Full Solutions

1. Express 2376 as a product of prime factors.

2376=2^3\times3^3\times11. Divide repeatedly by 2 and 3 until the remaining factor is prime.

2. Find the greatest integer that divides 450, 630 and 810 exactly.

450=2\times3^2\times5^2, 630=2\times3^2\times5\times7, 810=2\times3^4\times5. The common factors with smallest powers give 2\times3^2\times5=90.

3. Two machines complete cycles every 42 seconds and 70 seconds. How many times do they start together in the first 15 minutes, including the start?

\operatorname{LCM}(42,70)=210 seconds. Fifteen minutes is 900 seconds. Common starts occur at 0, 210, 420, 630 and 840 seconds, so there are 5.

4. Find the smallest number by which 540 must be multiplied to make a perfect square.

540=2^2\times3^3\times5. The unpaired factors are 3 and 5, so multiply by 15. Then 540\times15=2^2\times3^4\times5^2, a perfect square.

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