Prime Factors Hcf And Lcm

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Prime Factor Decomposition

Prime factors are prime numbers that multiply to give the original number (e.g., prime factors of 30 are 2, 3, 5).
Use a factor tree: split the number into factor pairs, continue until all branches end in primes, and circle the primes.
Write the product in ascending order, using index notation for repeated primes (e.g., 360 = 2³ × 3² × 5).
The decomposition is unique for each number (Fundamental Theorem of Arithmetic).
Common first primes: 2, 3, 5, 7, 11, 13, 17, 19.

Factor Tree for 360

Identifying Square and Cube Numbers

A number is a square number if all indices in its prime factor decomposition are even (e.g., 7056 = 2⁴ × 3² × 7²).
A number is a cube number if all indices are multiples of 3 (e.g., 1728000 = 2⁹ × 3³ × 5³).
To find the square root of a square number, halve all indices and multiply (e.g., √144 = √(2⁴×3²) = 2²×3 = 12).
For non-square numbers, rewrite with even indices where possible and simplify (e.g., √1440 = √(2⁴×3²×2×5) = 12√10).

Finding HCF Using Prime Factors

HCF (Highest Common Factor) is the largest number that divides both numbers exactly.
Method 1: List all factors of each number and pick the largest common one.
Method 2 (Venn diagram): Place common prime factors in the centre, multiply them to get HCF.
Method 3 (powers): For each common prime, take the smallest power appearing in both numbers, then multiply.
Example: HCF(36,120) = 2² × 3 = 12.

HCF of 36 and 120

Finding LCM Using Prime Factors

LCM (Lowest Common Multiple) is the smallest number that is a multiple of both numbers.
Method 1: List multiples of each number until a common one appears.
Method 2 (Venn diagram): Multiply all prime factors in the diagram (centre and outer regions).
Method 3 (powers): For every prime that appears, take the highest power from either number, then multiply.
Example: LCM(36,120) = 2³ × 3² × 5 = 360.

LCM of 36 and 120

Applications: Real-World Problems

Use LCM to find when events repeat simultaneously (e.g., trams every 9 and 12 min: LCM = 36 min, next at 9:36 am).
Use HCF to split items into equal groups (e.g., packets of 20 cheese slices and 12 burgers: HCF = 4, but for equal numbers use LCM = 60).
For problems with 'exactly the same number' of two items, find the LCM of the packet sizes.
For problems with 'next time together', find the LCM of the time intervals.

Working with Prime Factor Powers (A and B form)

When numbers are given as products of powers (e.g., A = 2³ × 3² × 5² × 11, B = 2⁴ × 3 × 5⁴ × 13), find HCF by taking minimum power of each common prime.
Find LCM by taking maximum power of each prime that appears in either number.
Example: HCF(A,B) = 2³ × 3 × 5² = 8×3×25=600; LCM = 2⁴ × 3² × 5⁴ × 11 × 13.

Finding Smallest Multiplier for a Square/Cube

To make a number a perfect square, multiply by primes to make all indices even.
To make a number a perfect cube, multiply by primes to make all indices multiples of 3.
Example: N = 2³ × 3² × 5⁷, to become a square multiply by 2 × 5 = 10 (so that indices become 4,2,8).

Examiner Tips

Always show clear working (factor tree or division) – marks awarded for method.
Write final answer in index form unless told otherwise.
HCF of two numbers can be one of the numbers (e.g., HCF(4,12)=4).
LCM of two numbers can be one of the numbers (e.g., LCM(4,12)=12).

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Câu hỏi luyện tập

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1.Which of the following is a prime number?
Easy
- A15
- B21
- C23
- D27
2.Express 84 as a product of its prime factors.
Easy
- A2² × 3 × 7
- B2 × 3 × 7
- C2² × 21
- D2³ × 3 × 7
3.Write 525 as a product of its prime factors.
Easy
- A3 × 5² × 7
- B3 × 5 × 7
- C5² × 21
- D3 × 5 × 35
4.A tram to Eccles leaves every 9 minutes and a tram to Didsbury leaves every 12 minutes. They both leave Piccadilly at 9 am. At what time will they next leave at the same time?
Medium
- A9:36 am
- B9:24 am
- C9:48 am
- D10:00 am
5.Find the highest common factor (HCF) of 90 and 48.
Medium
- A6
- B12
- C18
- D24
6.Write 56 as a product of its prime factors.
Easy
- A2³ × 7
- B2 × 28
- C2² × 14
- D2 × 4 × 7
7.Find the lowest common multiple (LCM) of 56 and 42.
Medium
- A168
- B336
- C84
- D252
8.A = 3² × 5⁴ × 7, B = 3⁴ × 5³ × 7 × 11. Find the HCF of A and B.
Medium
- A3² × 5³ × 7
- B3⁴ × 5⁴ × 7 × 11
- C3² × 5³
- D3⁴ × 5⁴ × 7

Thêm câu hỏi

9.A = 3⁵ × 5 × 7³, B = 2³ × 3 × 7⁴. Find the HCF of A and B.
Medium
- A3 × 7³
- B3⁵ × 7⁴
- C2³ × 3⁵ × 5 × 7⁴
- D3 × 7
10.Work out the lowest common multiple (LCM) of 36 and 120.
Medium
- A360
- B720
- C180
- D1080
11.Find the highest odd number that is a factor of 60 and a factor of 90.
Medium
- A15
- B5
- C3
- D45
12.Lamp A flashes every 20 seconds, Lamp B every 45 seconds and Lamp C every 120 seconds. They start flashing together. Not counting the start, how many times in the next hour will all three flash together at the same time?
Hard
- A30
- B20
- C10
- D60
13.Find the lowest common multiple (LCM) of 40 and 56.
Medium
- A280
- B560
- C140
- D1120
14.A = 2³ × 3 × 5, B = 2² × 3 × 5². Write down the highest common factor (HCF) of A and B.
Medium
- A2² × 3 × 5
- B2³ × 3 × 5²
- C2² × 3
- D2² × 5
15.Write 720 as a product of its prime factors.
Easy
- A2⁴ × 3² × 5
- B2³ × 3² × 5
- C2⁴ × 3 × 5
- D2⁵ × 3² × 5
16.Find the smallest whole number that 720 can be multiplied by to give a square number.
Hard
- A5
- B2
- C3
- D6
17.Express 180 as a product of its prime factors.
Easy
- A2² × 3² × 5
- B2³ × 3² × 5
- C2² × 3 × 5²
- D2 × 3² × 5²
18.Find the highest common factor (HCF) of 72 and 90.
Medium
- A6
- B9
- C12
- D18
19.Find the lowest common multiple (LCM) of 24 and 18.
Medium
- A36
- B48
- C72
- D96
20.A = 2³ × 3² × 5² × 11, B = 2⁴ × 3 × 5⁴ × 13. Find the lowest common multiple (LCM) of A and B.
Hard
- A2⁴ × 3² × 5⁴ × 11 × 13
- B2³ × 3 × 5²
- C2⁴ × 3² × 5⁴ × 11
- D2⁷ × 3³ × 5⁶ × 11 × 13
21.A machine has a buzzer that sounds every 50 minutes and a bell that sounds every 80 minutes. They sound together at 10 am. At what time do they next sound together?
Medium
- A4:40 pm
- B6:40 pm
- C8:40 pm
- D10:40 pm
22.Write 800 as a product of its prime factors.
Easy
- A2⁵ × 5²
- B2⁴ × 5³
- C2⁶ × 5²
- D2³ × 5⁴
23.Write 880 as a product of powers of its prime factors.
Easy
- A2⁴ × 5 × 11
- B2³ × 5 × 11²
- C2⁴ × 5² × 11
- D2³ × 5² × 11
24.Write 200 as a product of prime factors in index form.
Easy
- A2³ × 5²
- B2² × 5³
- C2⁴ × 5²
- D2³ × 5³
25.Find the highest common factor (HCF) of 75 and 105.
Medium
- A5
- B15
- C25
- D35
26.Write 36 as a product of prime factors. Give your answer in index form.
Easy
- A2² × 3²
- B2³ × 3²
- C2² × 3³
- D2 × 3²
27.Find the lowest common multiple (LCM) of 120 and 144.
Medium
- A720
- B1440
- C360
- D1080
28.234 = 2 × 3² × 13, 1872 = 2⁴ × 3² × 13. Use this information to write 438048 as a product of its prime factors.
Medium
- A2⁵ × 3⁴ × 13²
- B2⁵ × 3⁴ × 13
- C2⁴ × 3⁴ × 13²
- D2⁵ × 3² × 13²
29.Find the highest common factor (HCF) of 84 and 105.
Easy
- A7
- B21
- C3
- D42
30.A = 2³ × 5 × 7² × 11, B = 2⁴ × 7 × 11, C = 3 × 5². Find the lowest common multiple (LCM) of A, B and C.
Medium
- A2⁴ × 3 × 5² × 7² × 11
- B2⁴ × 3 × 5² × 7 × 11
- C2³ × 3 × 5² × 7² × 11
- D2⁴ × 3² × 5² × 7² × 11
31.The number 1 is a prime number.
Easy
True or false?
32.Which of the following is a factor of 24?
Easy
- A5
- B7
- C8
- D9
33.The prime factor decomposition of 144 is 2⁴ × 3². Which of the following is true?
Medium
- A144 is a square number
- B144 is a cube number
- C144 is both a square and a cube number
- D144 is neither a square nor a cube number
34.What is the highest common factor (HCF) of 24 and 36?
Easy
- A6
- B12
- C18
- D72
35.Match each number to its prime factor decomposition.
Medium
- 36
- 48
- 60
- 2²×3×5
- 2⁴×3
- 2²×3²
36.Arrange these numbers in increasing order of the number of distinct prime factors: 6, 30, 12.
Medium
- 6
- 12
- 30
37.Which of the following numbers are square numbers? Select all that apply.
Medium
- A2²×3⁴
- B2³×3²
- C2⁴×3²×5⁰
- D2²×3³
38.Match each number with its prime factor decomposition.
Medium
- 60
- 84
- 90
- 72
- 2² × 3 × 5
- 2² × 3 × 7
- 2 × 3² × 5
- 2³ × 3²
39.Arrange these numbers in increasing order of their highest common factor (HCF) with 60.
Hard
- 12
- 30
- 24
- 18
40.N = 2⁵ × 3⁴ × 7². Which of the following are factors of N? Select all that apply.
Hard
- A2³ × 3²
- B2⁵ × 3⁴ × 7
- C2⁴ × 3⁵
- D2² × 7³
- E2⁵ × 3² × 7
41.Complete the sentence.
Easy
A prime number has exactly ____ factors.
42.N = 2³ × 3² × 5⁷. Find the smallest positive integer A such that A×N is a perfect square.
Hard
43.Find the lowest common multiple (LCM) of 18 and 24.
Medium
44.Write down the first five prime numbers.
Easy
45.Find the lowest common multiple (LCM) of 28 and 42.
Medium
46.Find the smallest positive integer A such that A × N is a perfect square, where N = 2³ × 3² × 5⁷. Give your answer as an integer.
Hard
47.N = 2³ × 3² × 5⁷. Find the smallest positive integer A such that A×N is a perfect square.
Medium
- A9
- B11
- C10
- D12
48.Find the lowest common multiple (LCM) of 18 and 24.
Medium
- A65
- B72
- C79
- D86
49.Find the lowest common multiple (LCM) of 28 and 42.
Medium
- A76
- B92
- C84
- D100

Ghi chú ôn tập

## Prime Factors, HCF & LCM ### Prime Factor Decomposition - **Prime factor**: a prime number that divides a number exactly. - Use a **factor tree** to break a number into prime factors. - Write as product of primes in ascending order, using indices for repeated factors. - Example: 432 = 2⁴ × 3³. ### Uses of Prime Factor Decomposition - **Square number**: all indices are even. E.g., 7056 = 2⁴ × 3² × 7². - **Cube number**: all indices are multiples of 3. E.g., 1728000 = 2⁹ × 3³ × 5³. - **Square root**: halve the indices. E.g., √144 = √(2⁴×3²) = 2²×3 = 12. - **Exact square root**: rewrite odd indices as even+1, then separate. E.g., √1440 = √(2⁴×3²×2×5) = 2²×3×√10 = 12√10. ### Highest Common Factor (HCF) - **HCF**: largest number that divides both numbers. - **Method 1 (Venn diagram)**: write prime factors, put common ones in centre, multiply centre. - **Method 2 (powers)**: for each common prime, take the **lowest** power. Multiply. - Example: HCF(36,120): 36=2²×3², 120=2³×3×5 → common: 2²×3 = 12. ### Lowest Common Multiple (LCM) - **LCM**: smallest number that is a multiple of both numbers. - **Method 1 (Venn diagram)**: multiply all prime factors in the diagram. - **Method 2 (powers)**: for **every** prime appearing, take the **highest** power. Multiply. - Example: LCM(36,120): 36=2²×3², 120=2³×3×5 → highest powers: 2³×3²×5 = 360. ### Real-life Applications - **Scheduling**: find LCM of times to next coincide. E.g., trams every 9 and 12 min → LCM(9,12)=36 min → next at 9:36 am. - **Equal quantities**: find LCM of packet sizes. E.g., cheese slices (20) and burgers (12) → LCM(20,12)=60 → 3 packets of cheese, 5 boxes of burgers. ### Key Tips - Show factor tree clearly in exams. - HCF ≤ each number; LCM ≥ each number. - For numbers in index form, HCF = product of common primes with smallest indices; LCM = product of all primes with largest indices.

Prime factor: A prime number that divides a given number exactly.
Prime factor decomposition: Writing a number as a product of its prime factors, often using indices.
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.
Factor tree: A diagram used to break a number into prime factors by splitting into factor pairs.

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