The Lowest Common Multiple (LCM) of two or more numbers is the smallest multiple that is exactly divisible by each of the numbers. In other words, it is the smallest number that both (or all) of the numbers divide into evenly.
Key Concepts of LCM
1. Multiples
A multiple of a number is the product of that number and any other whole number. For example, the multiples of 4 are:
$$ 4, 8, 12, 16, 20, \dots $$
2. Common Multiples
The common multiples of two numbers are the numbers that appear in both of their lists of multiples. For example, the multiples of 4 and 6 are:
- Multiples of 4: $$ 4, 8, 12, 16, 20, \dots $$
- Multiples of 6: $$ 6, 12, 18, 24, 30, \dots $$
The common multiples are: $$ 12, 24, 36, \dots $$
3. Lowest Common Multiple (LCM)
The LCM is the smallest of the common multiples. For 4 and 6, the LCM is 12.
Methods for Finding the LCM
Method 1: Listing Multiples
You can find the LCM by listing the multiples of each number until you find the smallest one that appears in both lists.
Method 2: Prime Factorisation
This method involves expressing each number as a product of prime factors and taking the highest power of each prime factor involved.
For example, for 12 and 15:
- Prime factors of 12: $$ 2^2 \times 3 $$
- Prime factors of 15: $$ 3 \times 5 $$
The LCM is found by taking the highest powers of all prime factors:
$$ LCM = 2^2 \times 3 \times 5 = 60 $$
Practice Questions on LCM
Easy Level
- Find the LCM of 2 and 3.
- Find the LCM of 4 and 5.
- Find the LCM of 6 and 8.
- Find the LCM of 3 and 7.
- Find the LCM of 5 and 10.
- Find the LCM of 2 and 4.
- Find the LCM of 9 and 12.
- Find the LCM of 7 and 14.
- Find the LCM of 4 and 6.
- Find the LCM of 5 and 15.
- Find the LCM of 8 and 12.
- Find the LCM of 10 and 15.
- Find the LCM of 3 and 6.
- Find the LCM of 2 and 10.
- Find the LCM of 5 and 20.
- Find the LCM of 9 and 18.
- Find the LCM of 6 and 9.
- Find the LCM of 4 and 8.
- Find the LCM of 3 and 9.
- Find the LCM of 7 and 21.
Medium Level
- Find the LCM of 12 and 15.
- Find the LCM of 8 and 14.
- Find the LCM of 16 and 20.
- Find the LCM of 9 and 15.
- Find the LCM of 10 and 12.
- Find the LCM of 5, 10, and 15.
- Find the LCM of 6 and 14.
- Find the LCM of 12 and 18.
- Find the LCM of 7 and 11.
- Find the LCM of 4, 6, and 8.
- Find the LCM of 10 and 25.
- Find the LCM of 6, 9, and 12.
- Find the LCM of 18 and 24.
- Find the LCM of 8 and 16.
- Find the LCM of 15 and 20.
- Find the LCM of 12 and 30.
- Find the LCM of 14 and 21.
- Find the LCM of 6 and 15.
- Find the LCM of 9 and 21.
- Find the LCM of 8, 12, and 20.
Hard Level
- Find the LCM of 24 and 36.
- Find the LCM of 18 and 45.
- Find the LCM of 16, 20, and 30.
- Find the LCM of 25 and 40.
- Find the LCM of 30 and 45.
- Find the LCM of 32 and 48.
- Find the LCM of 20, 25, and 30.
- Find the LCM of 18, 24, and 30.
- Find the LCM of 9 and 27.
- Find the LCM of 15, 25, and 35.
- Find the LCM of 40 and 60.
- Find the LCM of 14, 28, and 42.
- Find the LCM of 50 and 75.
- Find the LCM of 22 and 44.
- Find the LCM of 16, 24, and 32.
- Find the LCM of 14 and 35.
- Find the LCM of 18, 36, and 54.
- Find the LCM of 33 and 66.
- Find the LCM of 25, 50, and 100.
- Find the LCM of 27 and 45.
Answers and Explanations
Easy Level
- LCM of 2 and 3 = 6
- Multiples of 2: $$ 2, 4, 6, 8, \dots $$
- Multiples of 3: $$ 3, 6, 9, 12, \dots $$
- LCM is 6.
- LCM of 4 and 5 = 20
- Multiples of 4: $$ 4, 8, 12, 16, 20, \dots $$
- Multiples of 5: $$ 5, 10, 15, 20, \dots $$
- LCM is 20.
- LCM of 6 and 8 = 24
- Multiples of 6: $$ 6, 12, 18, 24, 30, \dots $$
- Multiples of 8: $$ 8, 16, 24, 32, \dots $$
- LCM is 24.
- LCM of 3 and 7 = 21
- Multiples of 3: $$ 3, 6, 9, 12, 15, 18, 21, \dots $$
- Multiples of 7: $$ 7, 14, 21, 28, \dots $$
- LCM is 21.
- LCM of 5 and 10 = 10
- Multiples of 5: $$ 5, 10, 15, 20, \dots $$
- Multiples of 10: $$ 10, 20, 30, \dots $$
- LCM is 10.
- LCM of 2 and 4 = 4
- Multiples of 2: $$ 2, 4, 6, 8, 10, \dots $$
- Multiples of 4: $$ 4, 8, 12, \dots $$
- LCM is 4.
- LCM of 9 and 12 = 36
- Multiples of 9: $$ 9, 18, 27, 36, \dots $$
- Multiples of 12: $$ 12, 24, 36, \dots $$
- LCM is 36.
- LCM of 7 and 14 = 14
- Multiples of 7: $$ 7, 14, 21, \dots $$
- Multiples of 14: $$ 14, 28, \dots $$
- LCM is 14.
- LCM of 4 and 6 = 12
- Multiples of 4: $$ 4, 8, 12, 16, \dots $$
- Multiples of 6: $$ 6, 12, 18, 24, \dots $$
- LCM is 12.
- LCM of 5 and 15 = 15
- Multiples of 5: $$ 5, 10, 15, \dots $$
- Multiples of 15: $$ 15, 30, \dots $$
- LCM is 15.
Medium Level
- LCM of 12 and 15 = 60
- Prime factorisation of 12: $$ 2^2 \times 3 $$
- Prime factorisation of 15: $$ 3 \times 5 $$
- LCM = $$ 2^2 \times 3 \times 5 = 60 $$
- LCM of 8 and 14 = 56
- Prime factorisation of 8: $$ 2^3 $$
- Prime factorisation of 14: $$ 2 \times 7 $$
- LCM = $$ 2^3 \times 7 = 56 $$
- LCM of 16 and 20 = 80
- Prime factorisation of 16: $$ 2^4 $$
- Prime factorisation of 20: $$ 2^2 \times 5 $$
- LCM = $$ 2^4 \times 5 = 80 $$
- LCM of 9 and 15 = 45
- Prime factorisation of 9: $$ 3^2 $$
- Prime factorisation of 15: $$ 3 \times 5 $$
- LCM = $$ 3^2 \times 5 = 45 $$
- LCM of 10 and 12 = 60
- Prime factorisation of 10: $$ 2 \times 5 $$
- Prime factorisation of 12: $$ 2^2 \times 3 $$
- LCM = $$ 2^2 \times 3 \times 5 = 60 $$
Hard Level
- LCM of 24 and 36 = 72
- Prime factorisation of 24: $$ 2^3 \times 3 $$
- Prime factorisation of 36: $$ 2^2 \times 3^2 $$
- LCM = $$ 2^3 \times 3^2 = 72 $$
- LCM of 18 and 45 = 90
- Prime factorisation of 18: $$ 2 \times 3^2 $$
- Prime factorisation of 45: $$ 3^2 \times 5 $$
- LCM = $$ 2 \times 3^2 \times 5 = 90 $$
- LCM of 16, 20, and 30 = 240
- Prime factorisation of 16: $$ 2^4 $$
- Prime factorisation of 20: $$ 2^2 \times 5 $$
- Prime factorisation of 30: $$ 2 \times 3 \times 5 $$
- LCM = $$ 2^4 \times 3 \times 5 = 240 $$
- LCM of 25 and 40 = 200
- Prime factorisation of 25: $$ 5^2 $$
- Prime factorisation of 40: $$ 2^3 \times 5 $$
- LCM = $$ 2^3 \times 5^2 = 200 $$
These answers and explanations will help students to understand how to calculate the LCM using listing multiples and prime factorisation methods, tailored to Key Stage 3 level.
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