Prime Factors Hcf And Lcm

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Notes de leçon

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 = 2³ × 3² × 5).
Always show your factor tree clearly in exams to earn full marks.

Factor Tree for 360

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 (2²×3²) and 120 (2³×3×5) is 2²×3 = 12.

HCF of 36 and 120

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 (2²×3²) and 120 (2³×3×5) is 2³×3²×5 = 360.
The LCM can be one of the numbers (e.g., LCM of 4 and 12 is 12).

LCM of 36 and 120

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 2×2×2×2×3×3×3).
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.

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1.Write 220 as the product of its prime factors.
Easy
- A2² × 5 × 11
- B2 × 5 × 11
- C2² × 5 × 11²
- D2 × 5² × 11
2.Find the highest odd number that is a factor of 60 and a factor of 90.
Medium
- A15
- B30
- C45
- D5
3.Find the lowest common multiple (LCM) of 8 and 14.
Medium
- A56
- B112
- C28
- D42
4.Write 54 as a product of its prime factors.
Easy
- A2 × 3³
- B2² × 3³
- C2 × 3²
- D3³ × 2²
5.Write 18 as the product of its prime factors.
Easy
- A2 × 3²
- B2² × 3
- C2 × 3
- D3²
6.Find the lowest common multiple (LCM) of 15 and 27.
Medium
- A135
- B45
- C270
- D405
7.Write 825 as the product of its prime factors.
Medium
- A3 × 5² × 11
- B3 × 5 × 11²
- C5² × 11 × 3
- D3² × 5 × 11
8.Find the lowest common multiple (LCM) of 48 and 60.
Medium
- A240
- B120
- C480
- D360

Plus de questions

9.Write 195 as a product of its prime factors.
Easy
- A3 × 5 × 13
- B3 × 5 × 11
- C5 × 13 × 3
- D3 × 13 × 5
10.Write 60 as a product of its prime factors.
Easy
- A2² × 3 × 5
- B2 × 3 × 5
- C2² × 3² × 5
- D2 × 3² × 5
11.Find the lowest common multiple (LCM) of 36 and 60.
Medium
- A180
- B360
- C120
- D720
12.Write 432 as the product of its prime factors.
Medium
- A2⁴ × 3³
- B2³ × 3⁴
- C2⁴ × 3²
- D2² × 3⁴
13.Find the highest common factor (HCF) of 24 and 108.
Medium
- A12
- B6
- C24
- D18
14.Find the highest common factor (HCF) of 90 and 48.
Medium
- A6
- B12
- C3
- D24
15.Find the lowest common multiple (LCM) of 28 and 35.
Medium
- A140
- B280
- C70
- D420
16.Find the highest common factor of 48 and 80.
Medium
- A16
- B8
- C24
- D4
17.Express 84 as a product of its prime factors.
Easy
- A2² × 3 × 7
- B2 × 3 × 7
- C2² × 3² × 7
- D2 × 3² × 7
18.Find the highest common factor (HCF) of 18 and 30.
Easy
- A3
- B6
- C9
- D15
19.Find the lowest common multiple (LCM) of 6 and 10.
Easy
- A20
- B30
- C60
- D40
20.Write 180 as a product of its prime factors in index form.
Medium
- A2² × 3² × 5
- B2² × 3 × 5²
- C2 × 3² × 5²
- D2³ × 3² × 5
21.Find the highest common factor (HCF) of 56 and 84.
Medium
- A14
- B28
- C7
- D42
22.Find the lowest common multiple (LCM) of 12 and 18.
Medium
- A36
- B24
- C48
- D72
23.Write 200 as a product of its prime factors.
Medium
- A2³ × 5²
- B2² × 5³
- C2⁴ × 5
- D2 × 5⁴
24.Find the highest common factor (HCF) of 42 and 70.
Medium
- A7
- B14
- C21
- D35
25.Find the lowest common multiple (LCM) of 15 and 20.
Medium
- A60
- B30
- C75
- D90
26.The number N = 2³ × 3 × 5². Find the highest common factor (HCF) of N and 180.
Medium
- A60
- B20
- C30
- D90
27.Find the lowest common multiple (LCM) of 24 and 30.
Medium
- A120
- B60
- C240
- D180
28.Two buses arrive at a stop. One arrives every 15 minutes, the other every 20 minutes. At 09:00 they both arrive together. Find the next time they arrive together.
Medium
- A10:00
- B09:30
- C09:45
- D10:15
29.Find the highest common factor (HCF) of 126 and 180.
Medium
- A18
- B9
- C12
- D36
30.Find the lowest common multiple (LCM) of 28 and 98.
Medium
- A196
- B98
- C392
- D294
31.Gerry runs a lap in 90 seconds, Alain in 105 seconds. They start together. After how many minutes will they next be together at the start?
Medium
- A10.5 min
- B15 min
- C21 min
- D12 min
32.N = 2⁵ × 7². Find the highest common factor (HCF) of 168 and N.
Medium
- A56
- B84
- C28
- D42
33.Two numbers have prime factorisations: A = 2³ × 3 × 5 and B = 2² × 3² × 7. What is the highest common factor (HCF) of A and B?
Medium
- A12
- B24
- C36
- D48
34.Two buses arrive at a stop. One arrives every 18 minutes, the other every 24 minutes. At 10:47 they both arrive together. Find the next time they arrive together.
Medium
- A11:59
- B12:11
- C12:23
- D11:35
35.A rectangular garden has length 24 m and width 18 m. It is to be divided into identical square plots of the largest possible size. What is the side length of each square plot?
Medium
- A6 m
- B3 m
- C12 m
- D9 m
36.A number is written as 2³ × 3 × 5². What is the smallest integer by which this number must be multiplied to make a perfect square?
Hard
- A2 × 3 × 5
- B2 × 3
- C3 × 5
- D2 × 5
37.What is the smallest prime number?
Easy
- A0
- B1
- C2
- D3
38.Which of the following is a prime factor of 28?
Easy
- A4
- B6
- C7
- D14
39.Express 60 as a product of its prime factors.
Easy
- A2 × 3 × 5
- B2² × 3 × 5
- C2 × 3² × 5
- D2² × 3²
40.Which of the following is a prime number?
Easy
- A1
- B2
- C4
- D9
41.What is the highest common factor (HCF) of 12 and 18?
Easy
- A2
- B3
- C6
- D36
42.What is the lowest common multiple (LCM) of 4 and 6?
Easy
- A2
- B12
- C24
- D48
43.The number 1 is a prime number.
Easy
True or false?
44.A = 2³ × 3² × 5, B = 2² × 3 × 7. What is the highest common factor (HCF) of A and B?
Medium
- A2² × 3
- B2³ × 3² × 5 × 7
- C2² × 3² × 5 × 7
- D2³ × 3
45.A = 2³ × 3, B = 2² × 5. What is the lowest common multiple (LCM) of A and B?
Medium
- A2²
- B2³ × 3 × 5
- C2⁵ × 3 × 5
- D2³ × 5
46.Arrange the following numbers in increasing order of their highest common factor with 12: 6, 8, 9.
Medium
- 6
- 8
- 9
47.Match each pair of numbers to their highest common factor (HCF).
Medium
- 16 and 24
- 18 and 27
- 20 and 25
- 8
- 9
- 5
48.Which of the following are common factors of 24 and 36? (Select all that apply)
Medium
- A2
- B3
- C4
- D5
- E6
49.Match each number with its prime factor decomposition.
Medium
- 36
- 72
- 90
- 2³ × 3²
- 2² × 3²
- 2 × 3² × 5
50.Complete the sentence.
Easy
The prime factors of 30 are ____, ____, and ____.
51.Find the smallest positive integer that is a multiple of both 8 and 14.
Medium
52.Two numbers have prime factorisations: A = 2⁴ × 3² × 5 and B = 2² × 3³ × 7. What is the highest common factor (HCF) of A and B?
Medium
53.Two numbers have prime factorisations: A = 2⁴ × 3² × 5 and B = 2² × 3³ × 7. What is the highest common factor (HCF) of A and B?
Medium
- A36
- B44
- C40
- D32

Notes de révision

## 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).

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.

Cartes mémo

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 × 2 × 5 × 11 = 2² × 5 × 11.

Write 54 as a product of its prime factors.

54 = 2 × 3 × 3 × 3 = 2 × 3³.

Write 18 as a product of its prime factors.

18 = 2 × 3 × 3 = 2 × 3².

Write 825 as a product of its prime factors.

825 = 3 × 5 × 5 × 11 = 3 × 5² × 11.

Write 195 as a product of its prime factors.

195 = 3 × 5 × 13.

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

432 = 2⁴ × 3³.

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 = 2³×3, 108 = 2²×3³; common primes: 2²×3 = 12.)

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

HCF = 6. (90 = 2×3²×5, 48 = 2⁴×3; common: 2×3 = 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 = 2×3²×7, 180 = 2²×3²×5; common: 2×3² = 18.)

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

HCF = 14. (28 = 2²×7, 98 = 2×7²; common: 2×7 = 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 = 2³, 14 = 2×7; LCM = 2³×7 = 56.)

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

LCM = 135. (15 = 3×5, 27 = 3³; LCM = 3³×5 = 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 = 2²×3², 60 = 2²×3×5; LCM = 2²×3²×5 = 180.)

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

LCM = 140. (28 = 2²×7, 35 = 5×7; LCM = 2²×5×7 = 140.)

Find the lowest common multiple of 24 and 18.

LCM = 72. (24 = 2³×3, 18 = 2×3²; LCM = 2³×3² = 72.)

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

LCM = 60, HCF = 5. (20 = 2²×5, 15 = 3×5; LCM = 2²×3×5 = 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 ÷ 60 = 10.5 minutes.)

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