Properties of HCF and LCM: Rules, Formulas & Examples

To understand the key properties of HCF and LCM we first have to revise the definition and related concepts. Let's start with the definition followed by properties and examples.

Definition of HCF and LCM

The largest among the common factors of two or more given numbers is called the Highest Common Factor (HCF). It is also known as Greatest Common Divisor (GCD).

The smallest common multiple of two or more numbers is called the least common multiple or LCM of a number. 

List of HCF and LCM Properties 

Property 1: HCF for a set of numbers is always equal to or less than the given numbers.
Example: For 3, 6, 12, HCF is 3.
For 5, 6, 8, HCF is 1.

Property 2: HCF of two co-prime numbers is equal to 1.    
Example: For 3 and 8, HCF is 1.

Property 3: LCM for a set of numbers is always equal to or greater than the given numbers.
Example: For 5, 10, 15, LCM is 30
For 10, 20, 40, LCM is 40.

Property 4: LCM of two co-prime numbers is equal to their product.    
Example:  For 12, 13; LCM is 156.
For 15, 17; LCM is 255.

Property 5: HCF of a set of numbers is always a factor of their LCM.
Example: Consider the numbers 16 and 28.
Their prime factors are:
16 = 2 × 2 × 2 × 2
28 = 2 × 2 × 7
HCF is 4
LCM is 112
Clearly, we see that HCF 4 is a factor of LCM 112.

Property 6: In case, a number p is a factor of number q, then their HCF is p and LCM is q.
Example: Consider the numbers 9 and 18,
9 is a factor of 18.
HCF = 9; LCM = 18.

Property 7: Product of two numbers is equal to product of their LCM and HCF.
If p and q are two numbers, then,
Example: Consider the numbers 6 and 16
HCF = 2
LCM = 48
Now, using the formula we have, p × q = LCM × HCF
6 × 16 = 2 × 48 = 96

Solved Example on HCF and LCM

Examples1: Find the LCM of 6 and 8.
Solution:
Multiples of 6 = 6, 12, 18, 24, 30 and so on
Multiples of 8 = 8, 16, 24, 32, 40 and so on
LCM of 6 and 8 is 24.

Example 2: The LCM of two numbers is 50 and their product is 300. Find the HCF.
Solution: LCM of two numbers = 50
Product of the numbers = 300
We know that,
1st Number × 2nd Number = LCM × HCF
HCF = Product of the numbers/LCM
HCF = 300/50
HCF = 6

Example 3: 8 and 9 are two co-prime numbers. Using these numbers verify, LCM of Co-prime Numbers = Product Of The Numbers.
Solution: LCM and HCF of 8 and 9: 
8 = 2 × 2 × 2 = 2³ and 9 = 3 × 3 = 3²

LCM of 8 and 9 = 2³ × 3² = 8 × 9 = 72
HCF of 8 and 9 = 1
Product of 8 and 9 = 8 × 9 = 72

Hence, LCM of co-prime numbers = Product of the numbers. Therefore, verified.