What are LCM and HCF?
LCM (Least Common Multiple) is the smallest positive integer that is divisible by two or more numbers.
For example, the LCM of 4 and 6 is 12, because 12 is the smallest number that both 4 and 6 divide into without a remainder.
HCF (Highest Common Factor), also known as GCD (Greatest Common Divisor), is the largest positive integer that divides two or more numbers without leaving a remainder.
For example, the HCF of 12 and 18 is 6, because 6 is the largest number that divides both 12 and 18 evenly.
Methods to Calculate LCM and HCF
There are three primary methods used to find these values manually. This calculator uses the Prime Factorization method internally, which is the most robust for multiple numbers.
- Prime Factorization: Break each number down into its prime factors. The LCM is the product of the highest powers of all primes present. The HCF is the product of the lowest powers of common primes.
- Division Method: Divide the numbers by common prime factors until no more division is possible. Multiply the divisors to get the HCF. Multiply the divisors and remaining numbers to get the LCM.
- Euclidean Algorithm (for HCF): A repetitive process of division where the divisor becomes the dividend and the remainder becomes the divisor until the remainder is zero.
The Relationship Between LCM and HCF
For any two positive numbers, there is a fundamental relationship between their LCM, HCF, and the numbers themselves.
First Number × Second Number = LCM × HCF
a × b = LCM(a, b) × HCF(a, b)
This formula is very useful. If you know three of the values, you can easily find the fourth.
For instance, if the LCM of two numbers is 48 and their HCF is 4, and one of the numbers is 12, you can find the other number:
Other Number = (LCM × HCF) / First Number = (48 × 4) / 12 = 16.
Frequently Asked Questions
Can the LCM be smaller than the numbers?
No, the LCM is always greater than or equal to the largest number in the set. The only exception is if one of the numbers is zero (though LCM is usually defined for positive integers).
What is the HCF of prime numbers?
If you have two distinct prime numbers, their HCF will always be 1, because the only factor they share is 1. For example, the HCF of 7 and 13 is 1.
How is this used in real life?
LCM is used to find when events will happen at the same time (e.g., two buses arriving at the station simultaneously). HCF is used to simplify fractions or divide things into the largest possible equal groups (e.g., cutting a cloth into equal strips).