3. Least Common Multiple (LCM)
3. Least Common Multiple (LCM)
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Grasp the essence of the concept of LCM, recognizing it as the lowest common multiple
Identifying the Least Common Multiple of two or more numbers through prime factorization and/or the listing method.
Find the LCM of two or more numbers using different strategies
Explore the connections between the HCF and the Least Common Multiple (LCM)
The LCM of a set of two or more numbers is the smallest of their common multiples. It is the lowest number that is completely divisible by each of the given numbers.
Finding multiples:
Finding multiples:
To find the multiples of a number, multiply the number by 1, 2, 3 and so on.
Examples:
1. Find the first 3 multiples of 5
The first 3 multiples of 5: 5,10,15.
2. Is 25 a multiple of 6?
No, 25 is not a multiple of 6 as 6 x 4 = 24 and 6 x 5 = 30
3. Find the first 7 multiples of 7 that are
a. even
b. odd
1. Even multiples of 7: 14, 28, 42
2. Odd multiples of 7: 7, 21, 35 and 49
We can find LCM of two or more numbers by these methods:
1. Listing Method
2. Prime Factorization Method
1. Listing Method:
1. Listing Method:
Let us understand the steps of the listing method with an example given below.
Let us understand the steps of the listing method with an example given below.
Example 1: Find the LCM of 6 and 8.
Example 2: Find the LCM of 15 and 20 by listing method.
List the first several multiples of 15 and 20.
Multiples of 15:15,30,45,60,75,90,105,120, ….
Multiples of 20: 20,40,60,80,100,120,140,160, ….
Common multiples: 60, 120, …
Lowest Common multiple: 60
Watch a video to understand this better.
Vid 3.1 What do we mean by LCM?
2. Prime Factorization Method:
2. Prime Factorization Method:
To find the LCM using the prime factorization method we need to follow the below-mentioned steps. Let us solve an example to understand this better.
Example 1: Find the LCM of 8 and 24.
Example 2: Find the LCM of 12, 16 and 24.
Prime Factors of 12 = 2 x 2 x 3
Prime Factors of 16 = 2 x 2 x 2 x 2
Prime Factors of 24 = 2 x 2 x 2 x 3
For LCM of more than 2 numbers, we need to find the product of the common factors in all 3 numbers, then the common factors in 2 numbers and uncommon factors
LCM = Common Factors in 3 numbers x common factors in 2 numbers x Uncommon Factors
= (2 x 2) x (2 x 3) x 2
= 4 x 6 x 2
∴ LCM = 48
Example 3: Find LCM of 10, 18 and 20 by prime factorization method.
Prime Factors of 10: 2 x 5
Prime Factors of 18: 2 x 3 x 3
Prime Factors of 20: 2 x 2 x 5
LCM = Common Factors in 3 numbers x common factors in 2 numbers x Uncommon Factors
= (2) x (5) x (3 x 3 x 2)
∴ LCM = 180
Watch a video to understand this better.
Vid 3.2 LCM by Prime Factorisation Method
Extension
LCM and HCF are the two important methods in Maths. The LCM is used to find the least possible common multiples of two or more numbers whereas HCF is a method to find the highest possible common factor among the numbers. But both can be related by the formulae:
LCM(a,b) = a × b / HCF(a,b)
HCF(a,b) = a × b / LCM(a,b)
Where a and b are the two numbers.
Watch a video to understand this better.
Vid 3.3 Relation Between HCF, LCM & Given Numbers
Summary :-
The LCM of a set of two or more numbers is the smallest of their common multiples.
It is the lowest number that is completely divisible by each of the given numbers.
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