Page 47 - Maths Class 06

Lowest Common Multiple (LCM)
The Lowest Common Multiple (LCM) of two or more numbers is the smallest number which is a multiple
of each of the given numbers.
Consider the numbers 8 and 12.

Multiples of 8 = 8, 16, 24, 32, 40, 48,…
Multiples of 12 = 12, 24, 36, 48, 60,…

Common multiples of 8 and 12 are 24, 48,…
The smallest of the common multiples of 8 and 12 is 24.
Thus, LCM of 8 and 12 is 24.

In order to find the LCM of two or more numbers, we explain the procedure below through examples:
EX AM PLE 1. Find the LCM of 24 and 36.

SO LU TION : The prime factorizations of 24 and 36 are
24 = 2 2 2 3´ ´ ´
36 = 2 2 3 3´ ´ ´
In these prime factorizations,

2 occurs at most three times (in 24)
3 occurs at most two times (in 36)

So, the LCM of 24 and 36 is 2 2 2 3 3´ ´ ´ ´ , i e. . 72.
EX AM PLE 2. Find the LCM of 15, 60 and 105.
SO LU TION : The prime factorizations of 15, 60 and 105 are

15 = 3 5´
60 = 2 2 3 5´ ´ ´

105 = 3 5 7´ ´ .
In these prime factorizations,
2 occurs at most two times (in 60)

3 and 5 each occurs at most one time (in 15, 60 and 105)
7 occurs at most one time (in 105)
So, the LCM of 15, 60 and 105 is 2 2 3 5 7´ ´ ´ ´ , i e. . 420.

Some Prop er ties of HCF and LCM
1. The HCF of given numbers is not greater than any of the given numbers.

2. The HCF of two co-primes is 1.
3. The LCM of given numbers is not less than any of the given numbers.

4. The LCM of two co-primes is equal to their product.
5. The HCF of given numbers is always a factor of LCM.
6. If a and b are two given numbers such that a is a factor of b, then their HCF is a and LCM is b.

7. Product of two numbers = Product of their HCF and LCM.

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