Sec 1 Maths Chapter Guide

Prime Numbers, HCF and LCM

Learn prime numbers, prime factorisation, index notation, square roots, cube roots, Highest Common Factor and Lowest Common Multiple through clear explanations, visuals, worked examples and practice questions.

This chapter is written for Singapore Secondary 1 students, but parents and teachers can also use it to support revision and classroom teaching.

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Female teacher explaining prime numbers HCF and LCM to Sec 1 Maths students in a modern classroom

Chapter overview

In this chapter, students first revise factors and multiples, then learn how prime numbers are used to break composite numbers down. From there, students use index notation and prime factorisation to understand HCF and LCM more deeply.

Teachers can use this as a teaching sequence, students can use it as revision notes, and parents can use it to understand what their child is learning.

Start Factors and multiples

Build Prime numbers and factorisation

Apply HCF and LCM methods

Solve Word problems and practice

Learning section

Learning goals and prerequisites

By the end of this chapter, students should be able to explain the meaning of factors, multiples, prime numbers and composite numbers, and use them to solve HCF and LCM questions confidently.

Understand

Explain factors, multiples, prime numbers, composite numbers, square roots, cube roots, HCF and LCM in clear words.

Apply

Use listing, factor trees and prime factorisation to find HCF and LCM accurately.

Decide

Read word problems carefully and decide whether the situation requires HCF or LCM.

Prerequisite knowledge

Before starting this chapter, students should be comfortable with basic multiplication, division and whole numbers.

× tables division even / odd patterns word problems whole numbers

Key vocabulary

FactorDivides a number exactly.

MultipleMade by multiplying a number.

Prime numberHas exactly two factors.

Composite numberHas more than two factors.

HCFBiggest shared factor.

LCMSmallest shared multiple.

Concept 1

Factors and multiples

Factors and multiples are opposite ways of thinking about multiplication. A factor goes into a number exactly. A multiple is what we get after multiplying a number.

What is a factor?

A factor of a number is a whole number that divides it exactly without remainder.

1×12=12

2×6=12

3×4=12

Example: Find the factors of 12.

Factors of 12 are 1, 2, 3, 4, 6, 12.

A quick way to list factors is to write them in pairs. This helps students avoid missing a factor.

What is a multiple?

A multiple of a number is the result of multiplying that number by 1, 2, 3, 4 and so on.

4812162024

Example: List the first six multiples of 4.

4 × 1 = 4, 4 × 2 = 8, 4 × 3 = 12

First six multiples: 4, 8, 12, 16, 20, 24.

Multiples usually keep getting larger. Factors of a whole number are limited.

Checkpoint

Which list shows factors, and which list shows multiples?

List A

1, 2, 4, 8, 16

List B

8, 16, 24, 32, 40

Show answer

List A shows factors of 16. List B shows multiples of 8.

Concept 2

Prime and composite numbers

A prime number has exactly two factors. A composite number has more than two factors. This distinction matters because prime numbers are the building blocks of whole numbers.

Prime number

A prime number has exactly two factors: 1 and itself.

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Composite number

A composite number has more than two factors. It can be broken down into smaller factors.

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What about 1?

The number 1 is not prime and not composite. It has only one factor: itself.

Neither prime nor composite

How to test whether a number is prime

To test if a number is prime, check whether it is divisible by smaller prime numbers.

Example: Is 29 a prime number?

29 is not divisible by 2 because it is odd.

29 is not divisible by 3 because 2 + 9 = 11, which is not a multiple of 3.

29 is not divisible by 5 because it does not end with 0 or 5.

29 is a prime number.

Why 2 is special

The number 2 is the only even prime number.

2 has exactly two factors: 1 and 2.

Every other even number is divisible by 2 and has more than two factors.

For example, 8 has factors 1, 2, 4 and 8.

A common mistake is to say that all even numbers are composite. That is almost true, but 2 is the exception.

Concept 3

Prime factorisation

Prime factorisation means writing a composite number as a product of prime numbers. This helps students see what a number is made of.

Method 1: Factor tree

Start with the number. Split it into two factors. Continue splitting any composite number until only prime numbers are left.

12 = 2 × 6

6 = 2 × 3

So, 12 = 2 × 2 × 3

Using indices: 12 = 22 × 3

The final answer should only contain prime numbers. Since 6 is composite, we continue breaking it down.

Method 2: Repeated division

Divide the number by prime numbers step by step until the quotient becomes 1.

12 ÷ 2 = 6

6 ÷ 2 = 3

3 ÷ 3 = 1

So, 12 = 2 × 2 × 3

Using indices: 12 = 22 × 3

Repeated division is useful when numbers become larger because it keeps the working organised.

Another example: Prime factorise 18

18 = 2 × 9

9 = 3 × 3

So, 18 = 2 × 3 × 3

Using indices: 18 = 2 × 32

Checkpoint

Complete the prime factorisation of 20.

20 = 2 × 10

10 = 2 × 5

20 = ?

Show answer

20 = 2 × 2 × 5 = 22 × 5.

Concept 4

Index notation

Index notation is a shorter way to write repeated multiplication. In prime factorisation, it helps us write repeated prime factors neatly.

Index notation

23 × 32

What do the parts mean?

Base: The base is the number being multiplied repeatedly. In 23, the base is 2.

Index: The index tells us how many times the base is used as a factor. In 23, the index is 3.

Common readings

22 is read as two squared.

23 is read as two cubed.

52 means 5 × 5 = 25.

Students sometimes multiply the base by the index. For example, 23 is not 2 × 3. It means 2 × 2 × 2.

Example: Write 72 in index notation

72 = 8 × 9

8 = 2 × 4 and 4 = 2 × 2

9 = 3 × 3

So, 72 = 2 × 2 × 2 × 3 × 3

Using index notation: 72 = 23 × 32

Checkpoint

Write each expression using index notation.

3 × 3 × 3 × 5

2 × 2 × 7 × 7

5 × 5 × 5 × 5

Show answers

1. 33 × 5

2. 22 × 72

3. 54

Concept 5

Square roots and cube roots

Square roots and cube roots reverse square numbers and cube numbers. They help students connect repeated multiplication with the size of a square or cube.

Square root: finding the side of a square

3 × 3 = 9→√9 = 3

Square number: 9

Side length: 3

Square root: √9 = 3

√9 asks: “What number multiplied by itself gives 9?”

Cube root: finding the edge of a cube

3 × 3 × 3 = 27→∛27 = 3

Cube number: 27

Edge length: 3

Cube root: ∛27 = 3

∛27 asks: “What number multiplied by itself three times gives 27?”

Useful square numbers and cube numbers

22² = 4, 2³ = 8

33² = 9, 3³ = 27

44² = 16, 4³ = 64

55² = 25, 5³ = 125

Square roots and cube roots become easier when students recognise common square numbers and cube numbers quickly.

Checkpoint

Find the value of each expression.

√16

√25

∛8

∛64

Show answers

1. √16 = 4

2. √25 = 5

3. ∛8 = 2

4. ∛64 = 4

Concept 6

Highest Common Factor, HCF

The Highest Common Factor is the greatest factor shared by two or more numbers. It is useful when a question involves the largest equal group, the greatest possible size, or dividing items equally.

Finding HCF by listing factors

To find the HCF, list the factors of each number, identify the common factors, then choose the greatest one.

Factors of 12

1234612

Factors of 18

1236918

Highest common factorHCF = 6

Method steps

List the factorsWrite the factors of each number carefully.

Find the common factorsCommon factors appear in both factor lists.

Choose the greatest oneThe HCF is the largest number among the common factors.

HCF usually gives an answer that is smaller than or equal to the numbers given.

Worked example

Find the 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 factors: 1, 2, 3, 4, 6, 12

HCF of 24 and 36 = 12

Checkpoint

Find the HCF for each pair of numbers.

HCF of 10 and 15

HCF of 16 and 24

HCF of 18 and 30

Show answers

1. HCF of 10 and 15 = 5.

2. HCF of 16 and 24 = 8.

3. HCF of 18 and 30 = 6.

Concept 7

Lowest Common Multiple, LCM

The Lowest Common Multiple is the smallest multiple shared by two or more numbers. It is useful when a question involves repeated events, matching cycles, common timing, or the first time something happens together.

Finding LCM by listing multiples

To find the LCM, list the multiples of each number, identify the common multiples, then choose the smallest one.

Multiples of 4

4812162024

Multiples of 6

61218243036

Lowest common multipleLCM = 12

Method steps

List the multiplesWrite several multiples of each number.

Find the common multiplesCommon multiples appear in both multiple lists.

Choose the smallest oneThe LCM is the smallest number among the common multiples.

LCM usually gives an answer that is greater than or equal to the numbers given.

Worked example

Find the LCM of 8 and 12.

Multiples of 8: 8, 16, 24, 32, 40, 48

Multiples of 12: 12, 24, 36, 48, 60, 72

Common multiples include 24 and 48.

LCM of 8 and 12 = 24

Prime factorisation method

For larger numbers, prime factorisation can be a cleaner method.

8 = 23

12 = 22 × 3

LCM = 23 × 3 = 24

Checkpoint

Find the LCM for each pair of numbers.

LCM of 3 and 5

LCM of 6 and 9

LCM of 10 and 15

Show answers

1. LCM of 3 and 5 = 15.

2. LCM of 6 and 9 = 18.

3. LCM of 10 and 15 = 30.

Teacher note

Students often confuse HCF and LCM because both involve comparing two numbers. Keep the language very clear.

HCFLook for factors. The answer is the greatest shared factor.

LCMLook for multiples. The answer is the smallest shared multiple.

Concept 8

HCF vs LCM word problems

Many students can calculate HCF and LCM, but struggle to decide which one to use in a word problem. The key is to read what the question is really asking.

Decision guide

HCFUse when dividing things into the greatest possible equal groups.

LCMUse when finding the first common time, repeat, match, or meeting point.

Quick check: HCF answers are usually smaller. LCM answers are usually larger.

Clue words to look for

Largest, greatest, equal groups, same sizeThese often suggest HCF.

Again, together, same time, next, repeatThese often suggest LCM.

Do not rely on clue words alone. Students should still explain what is being grouped, repeated, divided, or matched.

Worked example 1: HCF

Question: A teacher has 24 pencils and 36 erasers. She wants to pack them into identical gift bags with no items left over. What is the greatest number of gift bags she can make?

The items are being divided into identical groups.

The question asks for the greatest number of groups.

HCF of 24 and 36 = 12. The teacher can make 12 gift bags.

Worked example 2: LCM

Question: A bell rings every 8 minutes and another bell rings every 12 minutes. If they ring together now, after how many minutes will they next ring together?

The bells are repeating at fixed intervals.

The question asks when they next happen together.

LCM of 8 and 12 = 24. The bells will ring together after 24 minutes.

Practice: choose HCF or LCM

Decide whether each problem needs HCF or LCM. Then solve it.

Two buses leave the interchange at the same time. One bus leaves every 10 minutes and another leaves every 15 minutes. When will they next leave together?

Mei has 18 red beads and 30 blue beads. She wants to make identical bracelets with no beads left over. What is the greatest number of bracelets she can make?

A light flashes every 6 seconds and another light flashes every 9 seconds. If both flash now, when will they next flash together?

Show answers

1. LCM. LCM of 10 and 15 = 30. They next leave together after 30 minutes.

2. HCF. HCF of 18 and 30 = 6. Mei can make 6 bracelets.

3. LCM. LCM of 6 and 9 = 18. They next flash together after 18 seconds.

Common mistake

A common mistake is to see two numbers and immediately calculate both HCF and LCM without understanding the situation.

Are we dividing into equal groups?If yes, the problem may need HCF.

Are repeated events meeting again?If yes, the problem may need LCM.

✓

Does the answer make sense?For HCF, the answer should divide the given numbers. For LCM, the given numbers should divide the answer.

Practice and review

Guided practice and chapter review

Use these questions to check whether students can connect factors, multiples, prime numbers, prime factorisation, square roots, cube roots, HCF and LCM.

Guided practice: core skills

List all the factors of 28.

Write the first five multiples of 7.

Decide whether 29 is prime or composite. Explain your answer.

Write 60 as a product of prime factors using index notation.

Show answers

1. Factors of 28: 1, 2, 4, 7, 14, 28.

2. First five multiples of 7: 7, 14, 21, 28, 35.

3. 29 is prime because it has only two factors: 1 and 29.

4. 60 = 2 × 2 × 3 × 5 = 22 × 3 × 5.

Guided practice: roots, HCF and LCM

Find √81.

Find ∛125.

Find the HCF of 18 and 42.

Find the LCM of 6 and 14.

Show answers

5. √81 = 9.

6. ∛125 = 5.

7. HCF of 18 and 42 = 6.

8. LCM of 6 and 14 = 42.

Mixed word problems

A teacher has 40 stickers and 56 bookmarks. She wants to pack them into identical sets with no items left over. What is the greatest number of sets she can make?

Two lights flash together now. One flashes every 12 seconds and the other flashes every 18 seconds. When will they next flash together?

Show answers

9. HCF question. HCF of 40 and 56 = 8. She can make 8 sets.

10. LCM question. LCM of 12 and 18 = 36. They next flash together after 36 seconds.

Chapter review checklist

FactorsNumbers that divide another number exactly.

MultiplesNumbers found by multiplying by 1, 2, 3 and so on.

Prime numbersNumbers with exactly two factors: 1 and itself.

Prime factorisationWriting a number as a product of prime factors.

HCFThe greatest factor shared by two or more numbers.

LCMThe smallest multiple shared by two or more numbers.

Teacher tip: ask students to explain why a word problem needs HCF or LCM before allowing them to calculate.

Teacher and parent support

Common mistakes, teaching notes and FAQs

This final section helps students avoid common errors, while giving teachers and parents a clearer way to support revision.

Common mistakes to watch for

Confusing factors and multiplesFactors divide a number exactly. Multiples are made by multiplying.

Treating 1 as a prime number1 has only one factor, so it is not prime.

Stopping prime factorisation too earlyContinue until every factor left is prime.

Mixing up HCF and LCMHCF is about common factors. LCM is about common multiples.

Teacher notes

Start with meaning before methodStudents should understand factors and multiples before HCF and LCM procedures.

Use visual comparison oftenFactor lists and multiple lists help students see the answer.

Ask for reasoning in word problemsAsk students whether the situation involves grouping, dividing, repeating or matching.

Parent notes

Parents do not need to reteach the whole chapter. The most useful support is to ask simple explanation questions and check whether the child can describe the method clearly.

Ask: “Why did you choose HCF or LCM?”This helps your child focus on understanding the story.

✓

Check divisibilityFor HCF, the answer should divide the numbers. For LCM, the numbers should divide the answer.

↺

Revise a little at a timeShort repeated practice is usually better than rushing many questions in one sitting.

Frequently asked questions

What is the difference between a factor and a multiple?

A factor divides a number exactly. A multiple is made when a number is multiplied by 1, 2, 3 and so on.

Is 1 a prime number?

No. A prime number has exactly two factors: 1 and itself. The number 1 has only one factor, so it is not prime.

When should students use HCF?

Students usually use HCF when a problem asks for the greatest possible equal group, equal size, or equal sharing with nothing left over.

When should students use LCM?

Students usually use LCM when a problem involves repeated events, common timing, cycles, or the first time two things happen together again.

Why do students struggle with HCF and LCM word problems?

Many students calculate correctly but choose the wrong method. They need to understand whether the story is about dividing into equal groups or finding a common repeated point.

Need help with Sec 1 Maths?

If your child understands the method but still struggles to apply it in word problems, start by revising the concept slowly before moving to mixed practice.

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