Prime Factors Hcf And Lcm

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Notes

Prime Factor Decomposition

A **prime number** has exactly two factors: itself and 1. First few primes: 2, 3, 5, 7, 11, 13, 17, 19, …
**Prime factors** are the prime numbers that multiply together to give the original number.
Use a **factor tree**: split the number into any pair of factors, then split each factor until all branches end in primes.
Circle the prime numbers at the ends of branches. The same primes appear regardless of the initial split.
Write the product of prime factors in ascending order, using **indices** for repeated primes $(e.g., 360 = 2^{3} \times 3^{2} \times 5)$ .
Always show your factor tree clearly in exams to earn full marks.

Finding HCF Using Prime Factors

The **Highest Common Factor (HCF)** is the largest number that divides both given numbers exactly.
Method 1 – Venn diagram: Write each number as a product of prime factors. Place common primes in the centre. Multiply the centre numbers to get the HCF.
Method 2 – Powers: For each common prime, take the **lowest power** that appears in both numbers. Multiply these together.
If no common prime factors exist, the HCF is 1.
Example: HCF of $36 (2^{2}\times 3^{2})$ and $120 (2^{3}\times 3\times 5)$ is $2^{2}\times 3 = 12$ .

Finding HCF by Listing Factors

List all factors of each number, then identify the largest factor appearing in both lists.
Factors of 36: 1, 2, 3, 4, 6, 9, 12, 18, 36. Factors of 120: 1, 2, 3, 4, 5, 6, 8, 10, 12, 15, 20, 24, 30, 40, 60, 120.
Common factors: 1, 2, 3, 4, 6, 12. The HCF is 12.
The HCF can be one of the numbers (e.g., HCF of 4 and 12 is 4).

Finding LCM Using Prime Factors

The **Lowest Common Multiple (LCM)** is the smallest number that is a multiple of both given numbers.
Method 1 – Venn diagram: Write each number as a product of prime factors. Place common primes in the centre. Multiply **all** numbers in the diagram (centre and outer regions).
Method 2 – Powers: For **every** prime that appears in either number, take the **highest power** that appears. Multiply these together.
Example: LCM of $36 (2^{2}\times 3^{2})$ and $120 (2^{3}\times 3\times 5)$ is $2^{3}\times 3^{2}\times 5 = 360$ .
The LCM can be one of the numbers (e.g., LCM of 4 and 12 is 12).

Finding LCM by Listing Multiples

List multiples of each number until a common multiple appears.
Multiples of 36: 36, 72, 108, 144, 180, 216, 252, 288, 324, 360, … Multiples of 120: 120, 240, 360, …
The smallest common multiple is 360.
This method is straightforward but can be time-consuming for large numbers.

Real-World Applications (LCM & HCF)

**LCM** is used to find when events that repeat at regular intervals coincide (e.g., bus schedules, runners on a track).
Example: Red bus every 18 min, blue bus every 24 min. LCM of 18 and 24 is 72 min. If they arrive together at 10:47, next together is $10:47 + 72$ $min = 11:59$ .
**HCF** is used to simplify fractions or divide items into equal groups.
Always check whether the problem requires HCF or LCM by reading carefully.

Exam Tips and Common Mistakes

Always show your factor tree or prime factor decomposition clearly.
Use indices to simplify the product of prime factors (e.g., 2⁴×3³ not $2\times 2\times 2\times 2\times 3\times 3\times 3)$ .
When using Venn diagrams, remember to include **all** prime factors from both numbers, not just common ones for LCM.
For HCF, only common primes are multiplied; for LCM, multiply every prime with the highest exponent.
Practice with past paper questions to become familiar with the style and wording.

Factor Tree for 432

Venn Diagram for HCF of 36 and 120

Venn Diagram for LCM of 36 and 120

Bus Schedule LCM Example

Practice questions

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1.Write 220 as the product of its prime factors.
Easy
- A $2^{2} \times 5 \times 11$
- B $2 \times 5 \times 11$
- C $2^{2} \times 5 \times 11^{2}$
- D $2 \times 5^{2} \times 11$
2.Find the highest odd number that is a factor of 60 and a factor of 90.
Easy
- A15
- B30
- C45
- D5
3.Find the lowest common multiple (LCM) of 8 and 14.
Easy
- A56
- B112
- C28
- D42
4.Write 54 as a product of its prime factors.
Easy
- A $2 \times 3^{3}$
- B $2^{2} \times 3^{3}$
- C $2 \times 3^{2}$
- D $3^{3} \times 2^{2}$
5.Write 18 as the product of its prime factors.
Easy
- A $2 \times 3^{2}$
- B $2^{2} \times 3$
- C $2 \times 3$
- D $3^{2}$
6.Find the lowest common multiple (LCM) of 15 and 27.
Easy
- A135
- B45
- C270
- D405
7.Write 825 as the product of its prime factors.
Medium
- A $3 \times 5^{2} \times 11$
- B $3 \times 5 \times 11^{2}$
- C $5^{2} \times 11 \times 3$
- D $3^{2} \times 5 \times 11$
8.Find the lowest common multiple (LCM) of 48 and 60.
Medium
- A240
- B120
- C480
- D360

Summary

## Prime Factors, HCF & LCM ### Prime Factor Decomposition - **Prime number**: exactly two factors (1 and itself). First primes: 2,3,5,7,11,13,17,19,… - **Prime factors**: primes that multiply to give the number. - **Method**: Use a factor tree. Split number into factor pairs; continue until all branches end in primes. Circle primes. Write product in ascending order, using powers for repeated factors. - Example: 432 = 2×2×2×2×3×3×3 = 2⁴×3³ ### Highest Common Factor (HCF) - **HCF**: largest number that divides both numbers exactly. - **Method 1 (Venn diagram)**: Write prime factors of each number. Place common primes in centre. HCF = product of centre primes. - **Method 2 (powers)**: For each common prime, take the smallest power. Multiply. - Example: HCF(36,120): 36=2²×3², 120=2³×3×5 → common primes: 2² and 3¹ → HCF = 2²×3 = 12. ### Lowest Common Multiple (LCM) - **LCM**: smallest number that is a multiple of both numbers. - **Method 1 (Venn diagram)**: Write prime factors. Place common primes in centre. LCM = product of all numbers in the circles. - **Method 2 (powers)**: For every prime that appears in either number, take the highest power. Multiply. - Example: LCM(36,120): 36=2²×3², 120=2³×3×5 → highest powers: 2³, 3², 5¹ → LCM = 2³×3²×5 = 360. ### Key Points - HCF can be one of the numbers (e.g., HCF(4,12)=4). - LCM can be one of the numbers (e.g., LCM(4,12)=12). - Use prime factor decomposition to solve problems involving bus/track times (find LCM).

Key terms

Prime factor decomposition: Writing a number as a product of its prime factors, often using powers for repeated primes.
Highest common factor (HCF): The largest number that divides two or more numbers exactly.
Lowest common multiple (LCM): The smallest number that is a multiple of two or more numbers.

Slides

Prime Factors

A prime number has exactly two factors: itself and 1.
Prime factors are prime numbers that multiply to give the original number.
Use a factor tree to break a number into prime factors.
Write the product in ascending order, using powers for repeated factors.

What is the prime factor decomposition of 54?

$2 \times 3^{3}$
$2^{2} \times 3^{2}$
$2 \times 3^{2}$
$2^{3} \times 3$

Highest Common Factor (HCF)

The HCF is the largest factor shared by two numbers.
Find it by listing factors or using prime factor Venn diagrams.
Multiply the common prime factors (lowest power of each common prime).

The HCF of 24 and 108 $is ____$ .

Lowest Common Multiple (LCM)

The LCM is the smallest multiple shared by two numbers.
Find it by listing multiples or using prime factor Venn diagrams.
Multiply all prime factors (highest power of each prime).

What is the LCM of 8 and 14?

Using Venn Diagrams for HCF & LCM

Write each number as a product of prime factors.
Place common primes in the centre of a Venn diagram.
$HCF =$ product of numbers in the centre.
$LCM =$ product of all numbers in the diagram.

Which of these are true?

The HCF of 42 and 90 is 6.
The LCM of 42 and 90 is 630.
The HCF of 36 and 120 is 18.

Using Powers of Prime Factors

For HCF, take the lowest power of each common prime.
For LCM, take the highest power of every prime that appears.
Multiply these powers together.

The LCM of 36 and 60 $is ____$ .

Real-World Applications

HCF and LCM are used in problems about timetables and events.
For example, two buses arriving at intervals: find LCM to know when they meet again.

A red bus arrives every 18 minutes, a blue bus every 24 minutes. They both arrive at 10:47. When is the next time they arrive together?

11:59
12:11
12:23
12:35

Key Points to Remember

Always show your factor tree or prime factor breakdown clearly.
Use indices to simplify your final answer.
The HCF can be one of the numbers; the LCM can also be one of the numbers.

Which statements are correct?

The HCF of 4 and 12 is 4.
The LCM of 4 and 12 is 12.
The HCF of 15 and 27 is 5.

Flashcards

What is a prime number?

A prime number has exactly two factors: itself and 1. Examples: 2, 3, 5, 7, 11, 13, 17, 19.

What are prime factors?

Prime factors are the prime numbers that multiply together to give the original number. For example, the prime factors of 30 are 2, 3, and 5.

How do you express 220 as a product of its prime factors?

$220 = 2 \times 2 \times 5 \times 11 = 2^{2} \times 5 \times 11$ .

Write 54 as a product of its prime factors.

$54 = 2 \times 3 \times 3 \times 3 = 2 \times 3^{3}$ .

Write 18 as a product of its prime factors.

$18 = 2 \times 3 \times 3 = 2 \times 3^{2}$ .

Write 825 as a product of its prime factors.

$825 = 3 \times 5 \times 5 \times 11 = 3 \times 5^{2} \times 11$ .

Write 195 as a product of its prime factors.

$195 = 3 \times 5 \times 13$ .

Write 432 as a product of powers of its prime factors.

$432 =$ 2⁴ $\times 3^{3}$ .

What is the highest common factor (HCF)?

The HCF of two numbers is the largest number that divides both exactly. For example, the HCF of 24 and 30 is 6.

Find the highest common factor (HCF) of 24 and 108.

$HCF = 12$ . $(24 = 2^{3}\times 3, 108 =$ 2²×3³; common primes: $2^{2}\times 3 = 12.)$

Find the highest common factor (HCF) of 90 and 48.

$HCF = 6$ . $(90 = 2\times 3^{2}\times 5, 48 =$ 2⁴×3; common: $2\times 3 = 6.)$

Find the highest common factor of 48 and 80.

$HCF = 16$ . $(48 =$ 2⁴×3, $80 =$ 2⁴×5; common: 2⁴ $= 16.)$

Find the highest common factor (HCF) of 126 and 180.

$HCF = 18$ . $(126 = 2\times 3^{2}\times 7, 180 =$ 2²×3²×5; common: $2\times 3^{2} = 18.)$

Find the highest common factor (HCF) of 28 and 98.

$HCF = 14$ . $(28 = 2^{2}\times 7, 98 =$ 2×7²; common: $2\times 7 = 14.)$

What is the lowest common multiple (LCM)?

The LCM of two numbers is the smallest number that is a multiple of both. For example, the LCM of 4 and 12 is 12.

Find the lowest common multiple (LCM) of 8 and 14.

$LCM = 56$ . $(8 = 2^{3}, 14 =$ 2×7; $LCM = 2^{3}\times 7 = 56.)$

Find the lowest common multiple (LCM) of 15 and 27.

$LCM = 135$ . $(15 = 3\times 5, 27 =$ 3³; $LCM = 3^{3}\times 5 = 135.)$

Find the lowest common multiple (LCM) of 48 and 60.

$LCM = 240$ . $(48 =$ 2⁴×3, $60 =$ 2²×3×5; $LCM =$ 2⁴×3×5 $= 240.)$

Find the lowest common multiple (LCM) of 36 and 60.

$LCM = 180$ . $(36 = 2^{2}\times 3^{2}, 60 =$ 2²×3×5; $LCM = 2^{2}\times 3^{2}\times 5 = 180.)$

Find the lowest common multiple (LCM) of 28 and 35.

$LCM = 140$ . $(28 = 2^{2}\times 7, 35 =$ 5×7; $LCM = 2^{2}\times 5\times 7 = 140.)$

Find the lowest common multiple of 24 and 18.

$LCM = 72$ . $(24 = 2^{3}\times 3, 18 =$ 2×3²; $LCM = 2^{3}\times 3^{2} = 72.)$

Find the lowest common multiple (LCM) and highest common factor (HCF) of 20 and 15.

$LCM = 60, HCF = 5$ . $(20 = 2^{2}\times 5, 15 =$ 3×5; $LCM = 2^{2}\times 3\times 5 = 60, HCF = 5.)$

Find the highest odd number that is a factor of 60 and a factor of 90.

15. (Factors of 60: 1,2,3,4,5,6,10,12,15,20,30,60; factors of 90: 1,2,3,5,6,9,10,15,18,30,45,90; highest odd common factor is 15.)

At a bus stop, a red bus arrives every 18 minutes and a blue bus every 24 minutes. At 10:47 both arrive. Find the next time they arrive together.

11:59. (LCM of 18 and 24 is 72 minutes; $10:47 + 72$ $min = 11:59.)$

Gerry takes 90 seconds per lap, Alain takes 105 seconds. They start together. After how many minutes are they next together?

10.5 minutes. (LCM of 90 and 105 is 630 seconds; $630 \div 60 = 10.5 minutes.)$

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