Prime Factors Hcf And Lcm

Learn it by playing

Answer these questions to earn energy, then fish and explore. No account needed.

For teachers: ready-to-use lesson slides, revision notes, diagrams for Prime Factors Hcf And Lcm (Maths [CIE], Extended) — use them in your lesson, or run the topic as a live class game.

Notes

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 $(e.g., 360 = 2^{3} \times 3^{2} \times 5)$ .
The decomposition is **unique** for each number (Fundamental Theorem of Arithmetic).
Common first primes: 2, 3, 5, 7, 11, 13, 17, 19.

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 $\times 3^{2} \times 7^{2})$ .
A number is a **cube number** if all indices are **multiples of 3** $(e.g., 1728000 =$ 2⁹ $\times 3^{3} \times 5^{3})$ .
To find the square root of a square number, **halve all indices** and multiply $(e.g., \sqrt{144} =$ √(2⁴×3²) $= 2^{2}\times 3 = 12)$ .
For non-square numbers, rewrite with even indices where possible and simplify $(e.g., \sqrt{1440} =$ √(2⁴×3²×2×5) $= 12\sqrt{10})$ .

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: $HCF(36,120) = 2^{2} \times 3 = 12$ .

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: $LCM(36,120) = 2^{3} \times 3^{2} \times 5 = 360$ .

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 $(e.g., A = 2^{3} \times 3^{2} \times 5^{2} \times 11, B =$ 2⁴ $\times 3 \times$ 5⁴ $\times 13)$ , 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: $HCF(A,B) = 2^{3} \times 3 \times 5^{2} =$ 8×3×25=600; $LCM =$ 2⁴ $\times 3^{2} \times$ 5⁴ $\times 11 \times 13$ .

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: $N = 2^{3} \times 3^{2} \times$ 5⁷, to become a square multiply by $2 \times 5 = 10 (so$ 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** $(e.g., LCM(4,12)=12)$ .

Factor Tree for 432

Venn Diagram for HCF and LCM of 42 and 90

Powers Method for HCF and LCM

Real-World LCM Example: Trams

Practice questions

Free preview — 8 of 32 questions. Sign up to see them all.

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
- A $2^{2} \times 3 \times 7$
- B $2 \times 3 \times 7$
- C $2^{2} \times 21$
- D $2^{3} \times 3 \times 7$
3.Write 525 as a product of its prime factors.
Easy
- A $3 \times 5^{2} \times 7$
- B $3 \times 5 \times 7$
- C $5^{2} \times 21$
- D $3 \times 5 \times 35$
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?
Easy
- 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.
Medium
- A $2^{3} \times 7$
- B $2 \times 28$
- C $2^{2} \times 14$
- D $2 \times 4 \times 7$
7.Find the lowest common multiple (LCM) of 56 and 42.
Medium
- A168
- B336
- C84
- D252
8. $A = 3^{2} \times 5^{4} \times 7, B = 3^{4} \times 5^{3} \times 7 \times 11$ . Find the HCF of A and B.
Medium
- A $3^{2} \times 5^{3} \times 7$
- B $3^{4} \times 5^{4} \times 7 \times 11$
- C $3^{2} \times 5^{3}$
- D $3^{4} \times 5^{4} \times 7$

Summary

## 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.

Key terms

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.

Slides

Prime Factor Decomposition

A prime number has exactly two factors: 1 and itself. The first few primes are 2, 3, 5, 7, 11, 13...
Prime factors are the prime numbers that multiply together to give the original number.
Use a factor tree: split the number into factors, then split each factor until all are prime.
Write the product in ascending order, using indices for repeated primes.

What is the prime factor decomposition of 84?

$2^{2} \times 3 \times 7$
$2 \times 3 \times 7$
$2^{2} \times 3^{2} \times 7$
$2^{3} \times 3 \times 7$

Using Prime Factor Decomposition

If all indices in the prime factor decomposition are even, the number is a square number.
If all indices are multiples of 3, the number is a cube number.
To find the square root of a square number, halve all the indices.
To find the exact square root of a non-square number, rewrite with even indices where possible.

The prime factor decomposition of 144 is 2⁴ $\times 3^{2}$ . Since all indices $are ____, 144$ is a square number. Its square root is $2^{2} \times 3 = ____$ .

Highest Common Factor (HCF)

The HCF of two numbers is the largest number that divides both exactly.
Method 1: List factors of each number and find the largest common one.
Method 2: Use prime factors: put common primes in a Venn diagram centre; multiply them.
Method 3: For numbers in index form, take the smallest power of each common prime.

What is the HCF of 36 and 120?

Lowest Common Multiple (LCM)

The LCM of two numbers is the smallest number that is a multiple of both.
Method 1: List multiples of each number and find the smallest common one.
Method 2: Use prime factors: multiply all primes in the Venn diagram (centre and outer).
Method 3: For numbers in index form, take the highest power of each prime that appears.

The LCM of 36 and 120 is 360. Using prime factors: $36 = 2^{2}\times 3^{2}, 120 = 2^{3}\times 3\times 5$ . Highest powers: $2^{3}, 3^{2}, 5$ . Product $= 8 \times 9 \times 5 = ____$ .

Real-Life Applications: Timings

Problems often involve events repeating at regular intervals (e.g., trams, buses, lamps).
The time when two events next coincide is the LCM of their intervals.
Example: A tram to Eccles every 9 min and to Didsbury every 12 min. They leave together at 9 am. Next time together is $LCM(9,12) = 36$ minutes later, at 9:36 am.

Buses to Acton leave every 24 minutes, to Barton every 20 minutes. Both leave at 9:00 am. When do they next leave together?

9:20 am
9:24 am
10:00 am
11:00 am

HCF and LCM with Index Form

When numbers are given as products of powers $(e.g., A = 2^{3}\times 3^{2}\times 5^{2}\times 11, B =$ 2⁴×3×5⁴×13), use the index method.
For HCF: take the smallest power of each common prime. For A and B, common primes: 2, 3, 5. Smallest powers: $2^{3}$ , 3¹, $5^{2}$ . $HCF = 2^{3}\times 3\times 5^{2}$ .
For LCM: take the highest power of every prime that appears. For A and B: 2⁴, $3^{2}$ , 5⁴, 11, 13. $LCM =$ 2⁴×3²×5⁴×11×13.

Given $A = 2^{3}\times 3^{2}\times 5^{2}\times 11$ and $B =$ 2⁴×3×5⁴×13, the HCF is $2^{3}\times 3\times 5^{2} = ____$ .

Finding Numbers from HCF and LCM

If you know the HCF and LCM of two numbers, you can find possible pairs.
The product of the two numbers equals $HCF \times LCM$ .
Both numbers must be multiples of the HCF, and their LCM must be a multiple of each number.

Martin's two numbers have $HCF = 6$ and LCM is a multiple of 15. Which pair is possible?

6 and 30
12 and 18
6 and 15
12 and 30

Square Numbers and Prime Factors

A number is a square if all prime factor indices are even.
To make a number into a square, multiply by the missing primes to make all indices even.
Example: $720 =$ 2⁴×3²×5. The index of 5 is 1 (odd), so multiply by 5 to get 2⁴×3²×5² $= (2^{2}\times 3\times 5)^{2} = 60^{2}$ .

$720 =$ 2⁴×3²×5. To make it a square, multiply $by ____$ . The smallest square number is 3600.

HCF and LCM of Three Numbers

For three numbers, the HCF is the product of the smallest power of each common prime across all three.
The LCM is the product of the highest power of each prime that appears in any of the numbers.
Use the same Venn diagram or index method extended to three sets.

Given $A=2^{3}\times 5\times 7^{2}\times 11$ , B=2⁴×7×11, $C=3\times 5^{2}$ , what is the LCM?

2⁴×3×5²×7²×11
$2^{3}\times 5\times 7\times 11$
2⁴×3×5×7²×11
2⁴×3×5²×7²×11

Summary

Prime factor decomposition: factor tree, write as product of primes with indices.
HCF: common primes with smallest powers (or Venn diagram centre).
LCM: all primes with highest powers (or all numbers in Venn diagram).
Applications: timing problems (LCM), finding numbers from HCF/LCM, square numbers.

Which of these are true?

The HCF of 12 and 18 is 6.
The LCM of 4 and 6 is 12.
A number with prime factor decomposition $2^{3}\times 3^{2}$ is a square number.
The LCM of two numbers is always greater than or equal to the HCF.

Unlock all 32 questions, slides & more

Create a free account to see every question, the slides, flashcards and revision notes for this topic.

Past papers

Past-paper practice for this topic is coming soon.

🗂️ Coming soon