Lowest common multiples and Highest common factors
🎯 In this topic you will
- Learn about lowest common multiples
- Learn about highest common factors
🧠 Key Words
- common factor
- common multiple
- digit
- factor
- highest common factor
- lowest common multiple
- multiple
Show Definitions
- common factor: A number that divides exactly into two or more numbers.
- common multiple: A number that is a multiple of two or more different numbers.
- digit: A single number from 0 to 9 used to make larger numbers.
- factor: A number that divides another number exactly, with no remainder.
- highest common factor: The largest number that is a factor of two or more numbers.
- lowest common multiple: The smallest number that is a multiple of two or more numbers.
- multiple: A number that results from multiplying a given number by a whole number.
🔁 Finding common multiples
The multiples of 4 are 4, 8, 12, 16, 20, 24, 28, 32, 36, 40, …
The multiples of 6 are 6, 12, 18, 24, 30, 36, 42, …
The common multiples of 4 and 6 are 12, 24, 36, …
The lowest common multiple (LCM) of 4 and 6 is 12.
🔎 Reasoning Tip
Common multiples: Multiply 4 by 1, 2, 3, and so on. The smallest number that is a multiple of both 4 and 6 is 12.
❓ EXERCISES
1. Write the first five multiples of:
- a) 5
- b) 10
- c) 7
- d) 12
👀 Show answer
b) 10, 20, 30, 40, 50
c) 7, 14, 21, 28, 35
d) 12, 24, 36, 48, 60
2.
- a) Write the multiples of 3 that are less than 40.
- b) Write the multiples of 5 that are less than 40.
- c) Find the common multiples of 3 and 5 that are less than 40.
👀 Show answer
b) 5, 10, 15, 20, 25, 30, 35
c) 15, 30
3.
- a) Find the common multiples of 4 and 3 that are less than 50.
- b) Complete this sentence: The common multiples of 4 and 3 are multiples of …
👀 Show answer
b) The common multiples of 4 and 3 are multiples of 12.
4. Find the LCM of 8 and 12.
5. Find the LCM of 10 and 15.
6. Find the LCM of 7 and 8.
👀 Show answer
5) LCM of 10 and 15 is 30
6) LCM of 7 and 8 is 56
🧠 Think like a Mathematician
Question: If $A$ and $B$ are two whole numbers, is $A \times B$ always a common multiple of $A$ and $B$? Is it also the lowest common multiple?
Equipment: Pencil, paper, calculator (optional)
Method:
- a) Show that the statement is true when $A = 4$ and $B = 7$.
- b) Show that the statement is true when $A = 6$ and $B = 5$.
- c) Is the statement always true? Give evidence to justify your answer.
- d) Consider this new statement:
If $A$ and $B$ are two whole numbers, then $A \times B$ is the lowest common multiple (LCM) of $A$ and $B$.
Is this statement true? Give evidence to justify your answer.
Follow-up Questions:
👀 Show Answers
- 1: Yes, $A \times B$ is always a common multiple of $A$ and $B$ because both numbers divide into it exactly.
- 2: No, it is not always the LCM. It is only the LCM when $A$ and $B$ have no common factors (i.e., they are coprime).
- 3: The LCM of two numbers is calculated as: $\text{LCM}(A, B) = \dfrac{A \times B}{\text{GCF}(A, B)}$
❓ EXERCISES
7. Find the LCM of 3, 4 and 6.
8. Find the LCM of 18, 9 and 4.
9. 21 is the LCM of two numbers. What are the numbers?
10. 30 is the LCM of two numbers. What are the numbers?
👀 Show answer
8) LCM of 18, 9, and 4 is 36
9) Possible numbers: 3 and 7 (LCM of 3 and 7 is 21)
10) Possible numbers: 5 and 6 or 10 and 3 (LCM of both pairs is 30)
🍬 Learning Bridge
After finding the lowest common multiple to work out shared multiples of numbers, you're now ready to flip the idea and look for the highest common factor — the biggest number that divides into both. These two skills work like opposites, but both help solve problems involving multiples, factors, and simplifying situations.
📏 Finding the HCF of two numbers
The factors of 18 are 1, 2, 3, 6, 9 and 18.
The factors of 27 are 1, 3, 9 and 27.
The common factors of 18 and 27 are 1, 3 and 9.
The highest common factor (HCF) of 18 and 27 is 9.
🔎 Reasoning Tip
Factor thinking: 18 can be written as 1 × 18, 2 × 9, or 3 × 6. The largest factor common to both 18 and 27 is 9.
📏 Finding the HCF of two numbers
The factors of 18 are 1, 2, 3, 6, 9 and 18.
The factors of 27 are 1, 3, 9 and 27.
The common factors of 18 and 27 are 1, 3 and 9.
The highest common factor (HCF) of 18 and 27 is 9.
❓ EXERCISES
11. Find the factors of:
- a) 24
- b) 50
- c) 45
- d) 19
12. Find the factors of:
- a) 33
- b) 34
- c) 35
- d) 36
- e) 37
13.
- a) Find the common factors of 18 and 48.
- b) Find the highest common factor of 18 and 48.
14. Find the highest common factor of:
- a) 12 and 28
- b) 12 and 30
- c) 12 and 36
15. Find the highest common factor of:
- a) 18 and 24
- b) 19 and 25
- c) 20 and 26
- d) 21 and 28
16. Find the highest common factor of:
- a) 60 and 70
- b) 60 and 80
- c) 60 and 90
17.
- a) Find the highest common factor of 35 and 56.
- b) Use your answer to part a to simplify the fraction $\dfrac{35}{56}$ as much as possible.
👀 Show answer
a) 1, 2, 3, 4, 6, 8, 12, 24
b) 1, 2, 5, 10, 25, 50
c) 1, 3, 5, 9, 15, 45
d) 1, 19
12)
a) 1, 3, 11, 33
b) 1, 2, 17, 34
c) 1, 5, 7, 35
d) 1, 2, 3, 4, 6, 9, 12, 18, 36
e) 1, 37
13)
a) 1, 2, 3, 6
b) 6
14)
a) 4
b) 6
c) 12
15)
a) 6
b) 1
c) 2
d) 7
16)
a) 10
b) 20
c) 30
17)
a) 7
b) $\dfrac{35 \div 7}{56 \div 7} = \dfrac{5}{8}$
18.
- a) Find the highest common factor of 25 and 36.
- b) Explain why the fraction $\dfrac{25}{36}$ cannot be simplified.
19. Find the highest common factor of 54, 72 and 90.
20. Two numbers have a highest common factor of 4. One of the numbers is between 10 and 20. The other number is between 20 and 40.
- a) What are the two numbers? Find all the possible answers.
- b) How can you be sure you have all the possible answers?
👀 Show answer
18b) The fraction $\dfrac{25}{36}$ cannot be simplified because 25 and 36 have no common factors other than 1.
19) HCF of 54, 72 and 90 is 18
20a)
Possible numbers between 10 and 20 that are multiples of 4: 12, 16, 20
Possible numbers between 20 and 40 that share HCF of 4 with one of the above: 20, 24, 28, 32, 36, 40
Valid pairs (HCF = 4):
• (12, 20), (12, 28), (12, 36)
• (16, 20), (16, 24), (16, 28), (16, 32), (16, 36), (16, 40)
• (20, 24), (20, 28), (20, 32), (20, 36), (20, 40)
20b)
Check all pairs of numbers (one in 10–20, the other in 20–40) whose HCF is exactly 4. List all such pairs systematically by checking each possible candidate.
🧠 Think like a Mathematician
Question: What is the relationship between the highest common factor (HCF), lowest common multiple (LCM), and the product of two numbers?
Equipment: Pencil, paper, calculator (optional)
Method:
- a) Find the HCF of 8 and 12.
- b) Find the LCM of 8 and 12.
- c) Find the product of 8 and 12.
- d) Find the product of the HCF and the LCM of 8 and 12.
- e) What do you notice about the answers to parts c and d?
- f) Can you generalise the result in part e for different pairs of numbers? Investigate.
Follow-up Questions:
👀 Show Answers
- 1:$8 \times 12 = 96$
- 2: HCF = 4, LCM = 24
- 3:$\text{HCF} \times \text{LCM} = 4 \times 24 = 96$, which matches the product of the original numbers.
- 4: Yes, for any two positive integers $A$ and $B$:
$A \times B = \text{HCF}(A,B) \times \text{LCM}(A,B)$
❓ EXERCISES
21. The HCF of two numbers is 3. The LCM of the two numbers is 45.
- a) Explain why each number is a multiple of 3.
- b) Explain why each number is a factor of 45.
- c) Find the two numbers.
👀 Show answer
b) Since the LCM is 45, both numbers must be factors of 45 — otherwise their least common multiple would be higher.
c) The two numbers are 9 and 15, because:
– HCF(9, 15) = 3
– LCM(9, 15) = 45
⚠️ Be careful!
Don’t confuse the lowest common multiple (LCM) with the highest common factor (HCF). The LCM is the smallest number that both numbers divide into, while the HCF is the largest number that divides into both.