Least Common Multiple (L.C.M.)
The multiples which are common to a given set of numbers are called their common multiples.
For example, the common multiples of 6 and 8 are 24, 48, 72, 96 and so on.
The lowest among these common multiples is 24 which is the Lowest Common Multiple of 6 and 8.
Thus, the multiple which is smallest among the common multiples of a given set of numbers is called their Lowest Common Multiple (L.C.M.).
Example- Let us take 2 numbers 12 and 18 and find their LCM.
Multiples of 12 = 12, 24, 36, 48, 60, 72, 84, 96, 108, …
Multiples of 18 = 18, 36, 54, 72, 90, 108, 126, …
Common multiples of 12 and 18 are 36, 72, 108 and so on.
Smallest among all common multiples is 36.
Thus, L.C.M. of 12 and 18 = 36
- LCM byPrime Factorization Method
- In this method, we express the given set of numbers as a product of their prime factors.
- The LCM of the twonumbers is calculated as the product of the prime factors counted the maximum numberof times they occur in any of the numbers.
Example-
- Find the LCM of 18 and 24.
Let us find the prime factorization of 18 and 24.
So, 18 = 2 × 3 × 3
24 = 2 × 2 × 2 × 3
Maximum number of times prime factor 2 occurs among 18 and 24 is 3.
Similarly, 3 occurs maximum of 2 times.
LCM of 18 and 24 = 2 × 2 × 2 × 3 × 3 = 72
Thus, LCM (18 and 24) = 72
Relationship between HCF and LCM
The product of HCF and LCM of two numbers is equal to the product of the numbers.
i.e. HCF × LCM = Product of the Numbers
Example- Given two numbers: 6 and 8
HCF (6 and 8) = 2
LCM (6 and 8) = 24
Product (6 and 8) = 6 × 8 = 48
Now, HCF × LCM = 2 × 24 = 48
Thus, HCF × LCM = Product of the Numbers.
Can you solve this??
#Find HCF and LCM of the form- 2222….30 times and 3333….70 times (any random numbers)
Solution:
HCF
- Take HCF of no. that is being repeated…i.e. HCF of 2 & 3.
i.e. 1 - Take HCF of no. of time these nos. are being repeated i.e. HCF of 30 and 70 is 10.
- So, the HCF of the two given numbers is 1…written 10 times.
LCM
- Take LCM of no. that is being repeated…i.e. LCM of 2 & 3.
i.e. 6 - Take LCM of no. of time these nos. are being repeated i.e. LCM of 30 and 70 is 210.
- So, the LCM is 6…written 210 times
