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🔢 Real Numbers • Lesson 3 of 29

Highest Common Factor and Lowest Common Multiple

Learn how to find the largest factor shared by numbers, the smallest multiple they share, and how to decide which method a real-life problem requires.

🔢 Real Numbers ✏️ Worked examples included 🧠 Step-by-step learning

🎬 See common multiples meet

The first shared multiple is the lowest common multiple.

🎯 By the end of this lesson...

You should be able to:

🤔 Think About This...

🎀

Cutting ribbon

A decorator has one ribbon measuring 24 cm and another measuring 36 cm.

She wants to cut both ribbons into equal pieces that are as long as possible, without wasting any ribbon.

What should the length of each piece be?

This problem asks for the largest length that divides both measurements exactly.

We need the Highest Common Factor.
🚌

Bus departures

One bus leaves a terminus every 12 minutes. Another leaves every 18 minutes.

They leave together at 8:00 a.m. After how many minutes will they next leave together?

This problem asks when two repeating events will happen together again.

We need the Lowest Common Multiple.

1️⃣ Common Factors

A common factor is a number that divides two or more numbers exactly.

Example 1: Find the common factors of 12 and 18

List the factors of each number.

Factors of 12

1, 2, 3, 4, 6, 12

Factors of 18

1, 2, 3, 6, 9, 18

Look for numbers that appear in both lists.

Common factors: 1, 2, 3 and 6

2️⃣ Highest Common Factor

The Highest Common Factor, written as HCF, is the largest factor shared by two or more numbers.

Common factors of 12 and 18:

1, 2, 3, 6

The highest number in the list is 6.

HCF of 12 and 18 = 6
Some books use the term Greatest Common Factor, abbreviated as GCF. It means the same thing as HCF.

🧠 Method 1: Finding HCF by Listing Factors

Follow these steps:

Step 1: List all the factors of each number.

Step 2: Identify the factors that appear in every list.

Step 3: Choose the largest common factor.

Example 2: 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 and 12
HCF of 24 and 36 = 12

🌍 Real-Life HCF Example

Example 3: Making identical stationery packs

A teacher has:

  • 24 pencils;
  • 36 rulers.

She wants to make the greatest possible number of identical packs without leaving anything over.

How many packs can she make?

The words greatest possible number of identical packs tell us to use HCF.
HCF of 24 and 36 = 12

Therefore, she can make 12 identical packs.

Pencils in each pack:

24 ÷ 12 = 2
Rulers in each pack:

36 ÷ 12 = 3
She can make 12 packs, with 2 pencils and 3 rulers in each pack.

3️⃣ Common Multiples

A common multiple is a number that appears in the multiples lists of two or more numbers.

Example 4: Find common multiples of 4 and 6

Multiples of 4

4, 8, 12, 16, 20, 24, 28, 32, 36, …

Multiples of 6

6, 12, 18, 24, 30, 36, 42, …

Common multiples:

12, 24, 36, …
Two numbers have infinitely many common multiples.

4️⃣ Lowest Common Multiple

The Lowest Common Multiple, written as LCM, is the smallest positive multiple shared by two or more numbers.

Common multiples of 4 and 6:

12, 24, 36, …

The smallest positive common multiple is 12.

LCM of 4 and 6 = 12

🧠 Method 1: Finding LCM by Listing Multiples

Follow these steps:

Step 1: List positive multiples of each number.

Step 2: Continue until the same number appears in every list.

Step 3: Choose the smallest positive common multiple.

Example 5: Find the LCM of 8 and 12

Multiples of 8

8, 16, 24, 32, 40, 48, …

Multiples of 12

12, 24, 36, 48, …

The first common multiple is 24.

LCM of 8 and 12 = 24

🌍 Real-Life LCM Example

Example 6: School bell reminders

During a school event:

  • one reminder bell rings every 15 minutes;
  • another reminder rings every 20 minutes.

Both bells ring together at 9:00 a.m. When will they next ring together?

The words next happen together tell us to use LCM.

Multiples of 15

15, 30, 45, 60, …

Multiples of 20

20, 40, 60, …

LCM of 15 and 20 = 60

The bells will ring together again after 60 minutes.

They will next ring together at 10:00 a.m.

5️⃣ Using Prime Factorisation

Listing works well for small numbers.

For larger numbers, prime factorisation is often quicker and more reliable.

Write each number as a product of prime factors before finding its HCF or LCM.

🧠 Finding HCF Using Prime Factors

StudyNest Method

Step 1: Write each number as a product of prime factors.

Step 2: Identify prime factors shared by every number.

Step 3: For each shared prime, use the smallest power that appears.

Step 4: Multiply those shared prime factors.

Example 7: Find the HCF of 72 and 120

72 = 2³ × 3²
120 = 2³ × 3 × 5

The shared prime factors are 2 and 3.

For the factor 2

Both numbers contain 2³.

Use 2³

For the factor 3

The powers are 3² and 3¹.

Use the smaller power: 3¹
HCF = 2³ × 3
HCF = 8 × 3
HCF of 72 and 120 = 24

🧠 Finding LCM Using Prime Factors

StudyNest Method

Step 1: Write each number as a product of prime factors.

Step 2: Collect every different prime factor that appears.

Step 3: For each prime, use the greatest power that appears.

Step 4: Multiply all the selected prime powers.

Example 8: Find the LCM of 72 and 120

72 = 2³ × 3²
120 = 2³ × 3 × 5

We need every prime factor appearing in either number.

Factor 2

Greatest power: 2³

Factor 3

Greatest power: 3²

Factor 5

Greatest power: 5¹

LCM = 2³ × 3² × 5
LCM = 8 × 9 × 5
LCM of 72 and 120 = 360

🔑 Smallest for HCF, Greatest for LCM

HCF

Use only the prime factors shared by all the numbers.

Choose the smallest power of each shared prime.

LCM

Use every prime factor that appears in any of the numbers.

Choose the greatest power of each prime.

HCF: shared primes with the smallest powers.

LCM: all required primes with the greatest powers.

6️⃣ Choosing HCF or LCM

Many students can calculate HCF and LCM but struggle to decide which one a word problem requires.

Think HCF when...

  • cutting into the longest equal pieces;
  • making the greatest number of identical groups;
  • finding the largest tile or square that fits exactly;
  • sharing everything equally with nothing left over;
  • looking for the greatest possible size.

Think LCM when...

  • events must happen together again;
  • cycles or schedules repeat;
  • finding the smallest quantity divisible by several numbers;
  • working with repeated bus, bell or machine intervals;
  • looking for the earliest shared time.

🧠 StudyNest Word-Problem Method

Step 1: Identify what the question is asking you to find.

Step 2: Look for the underlying idea, not only one keyword.

Step 3: Ask:

“Am I dividing quantities into the greatest equal groups or pieces?”

If yes, try HCF.

Step 4: Ask:

“Am I looking for the first time or quantity at which repeating patterns meet?”

If yes, try LCM.

Step 5: Calculate and write the answer in the context of the question.

Step 6: Check whether the answer makes sense.
Do not choose a method from one word alone. Read the complete situation and identify what is happening.

🌍 More Real-Life Examples

Example 9: Tiling a rectangular floor

A rectangular floor measures 360 cm by 240 cm.

It must be covered using the largest possible square tiles without cutting any tile.

Find the side length of each tile.

We need the largest measurement that divides both dimensions exactly, so we use HCF.
360 = 2³ × 3² × 5
240 = 2⁴ × 3 × 5
HCF = 2³ × 3 × 5
HCF = 120
Each square tile should have a side length of 120 cm.

Example 10: Medicine reminders

A patient must take:

  • one medicine every 6 hours;
  • another medicine every 8 hours.

Both medicines are taken together at 6:00 a.m. After how many hours will they next be taken together?

The schedules repeat and must meet again, so we use LCM.
Multiples of 6:

6, 12, 18, 24, …
Multiples of 8:

8, 16, 24, …
LCM of 6 and 8 = 24
The medicines will next be taken together after 24 hours, at 6:00 a.m. the following day.

Example 11: Organising sports teams

A school has 48 boys and 60 girls taking part in a sports programme.

The organisers want to divide them into the greatest possible number of identical mixed teams, with no learner left out.

How many teams can be formed?

HCF of 48 and 60 = 12
Boys per team:

48 ÷ 12 = 4
Girls per team:

60 ÷ 12 = 5
The school can form 12 identical teams, each containing 4 boys and 5 girls.

✅ Checking Your Answers

Check an HCF

The HCF must divide every original number exactly.

24 ÷ 12 = 2
36 ÷ 12 = 3

Check an LCM

The LCM must be divisible by every original number.

24 ÷ 8 = 3
24 ÷ 12 = 2

⚠ Common Mistakes

Mistake 1:

Choosing LCM for a problem about cutting quantities into the largest equal pieces.

Greatest equal pieces normally require HCF.
Mistake 2:

Choosing HCF when two repeating schedules must meet again.

Repeating cycles normally require LCM.
Mistake 3:

Finding common factors but forgetting to choose the highest one.
Mistake 4:

Finding a common multiple but not checking that it is the lowest positive one.
Mistake 5:

Using the greatest powers when calculating HCF.

HCF uses the smallest powers of the shared primes.
Mistake 6:

Using only the shared prime factors when calculating LCM.

LCM must include every prime factor needed by either number.
Mistake 7:

Giving only a number without answering the real-life question.

Always include units, time or the meaning of the answer.

🎮 Your Turn!

  1. Find the HCF of 18 and 30 by listing their factors.
  2. Find the HCF of 28 and 42.
  3. Find the LCM of 6 and 9 by listing multiples.
  4. Find the LCM of 8 and 14.
  5. Given that 48 = 2⁴ × 3 and 72 = 2³ × 3², find their HCF.
  6. Given that 48 = 2⁴ × 3 and 72 = 2³ × 3², find their LCM.
  7. Find the HCF and LCM of 36 and 90 using prime factorisation.
  8. A shop has 30 bottles of water and 45 cartons of juice. The owner wants to make the greatest possible number of identical refreshment packs with nothing left over. How many packs can be made?
  9. One security light flashes every 10 seconds and another flashes every 15 seconds. They flash together now. After how many seconds will they next flash together?
  10. Two pieces of fabric measure 84 cm and 126 cm. They must be cut into equal pieces of the greatest possible length. Find the length of each piece.
  11. A school club meets every 12 days and a sports club meets every 18 days. Both clubs meet today. After how many days will they next meet on the same day?
  12. A rectangular board measures 150 cm by 210 cm. It is to be divided into the largest possible identical square sections. Find the side length of each square.
✅ Show Answers
  1. HCF = 6
  2. HCF = 14
  3. LCM = 18
  4. LCM = 56
  5. HCF = 2³ × 3 = 24
  6. LCM = 2⁴ × 3² = 144
  7. 36 = 2² × 3² and 90 = 2 × 3² × 5

    HCF = 2 × 3² = 18

    LCM = 2² × 3² × 5 = 180
  8. HCF of 30 and 45 = 15. Therefore, 15 identical packs can be made.
  9. LCM of 10 and 15 = 30. They will next flash together after 30 seconds.
  10. HCF of 84 and 126 = 42. Each piece should be 42 cm long.
  11. LCM of 12 and 18 = 36. They will next meet on the same day after 36 days.
  12. HCF of 150 and 210 = 30. Each square should have a side length of 30 cm.

🧠 Remember This

HCF is the greatest factor shared by the numbers.

LCM is the smallest positive multiple shared by the numbers.

HCF often appears in cutting, grouping and sharing problems.

LCM often appears in repeating schedules and cycles.

For prime factors:

HCF → shared primes, smallest powers

LCM → all required primes, greatest powers

🎯 Before Moving On...

🎉 Well Done!

You can now find HCF and LCM and apply them to practical situations.

Remember:

Greatest equal groups or pieces → HCF

First time repeating patterns meet → LCM

📈 Your Progress

Lesson 3 of 17

Real Numbers Progress 18%