LCM (Least Common Multiple) and HCF (Highest Common Factor)
Definition:
The LCM of two or more integers is the smallest positive integer that is divisible by each of the integers.
The HCF of two or more integers is the largest positive integer that divides each of the integers without leaving a remainder.
Basic Formulas:
- LCM(a, b) * HCF(a, b) = a * b
- LCM(a, b) = (a * b) / HCF(a, b)
Trick for Finding LCM and HCF:
- To find LCM: List down the multiples of each number and identify the smallest common multiple.
- To find HCF: Use prime factorization or division method to find the common factors.
Examples:
Example 1:
Find the LCM and HCF of 12 and 18.
Solution:
For LCM: Multiples of 12 are 12, 24, 36, ... and multiples of 18 are 18, 36, 54, ... The smallest common multiple is 36. So, LCM(12, 18) = 36.
For HCF: Prime factorization of 12 is 2 * 2 * 3 and of 18 is 2 * 3 * 3. Common factors are 2 and 3. So, HCF(12, 18) = 6.