# Properties of HCF and LCM

Properties of HCF and LCM: For the better understanding of the concept LCM (Lowest Common Multiple) and HCF (Highest Common Factor), we need to recollect the terms multiples and factors. Let’s learn about LCM, HCF, and relation between HCF and LCM of natural numbers.

Learn in detail: Hcf And Lcm

## Definition of LCM and HCF

Lowest Common Multiple (LCM):  The least or smallest common multiple of any two or more given natural numbers are termed as LCM. For example, LCM of 10, 15, and 20 is 60.

Highest Common Factor (HCF): The largest or greatest factor common to any two or more given natural numbers is termed as HCF of given numbers. Also known as GCD (Greatest Common Divisor). For example, HCF of 4, 6 and 8 is 2.

4 = 2 × 2

6 =3 × 2

8 = 4 × 2

Here, the highest common factor of 4, 6 and 8 is 2.

Both HCF and LCM of given numbers can be found by using two methods; they are division method and prime factorization.

## HCF and LCM Formulas

Property 1: The product of LCM and HCF of any two given natural numbers is equivalent to the product of the given numbers.

LCM × HCF = Product of the Numbers

Suppose A and B are two numbers, then.

LCM (A & B) × HCF (A & B) = A × B

Property 2: HCF of co-prime numbers is 1. Therefore LCM of given co-prime numbers is equal to the product of the numbers.

LCM of Co-prime Numbers = Product Of The Numbers

Property 3: H.C.F. and L.C.M. of Fractions

LCM of fractions = $\frac{LCM \: of\: numerators}{HCF \: of\: denominators}$

HCF of fractions = $\frac{HCF \: of\: numerators}{LCM \: of\: denominators}$

## HCF and LCM Problems

Example 1: Prove that: LCM (9 & 12) × HCF (9 & 12) = Product of 9 and 12

Solution: LCM and HCF of 9 and 12:

9 = 3 × 3 = 3²

12 = 2 × 2 × 3 = 2² × 3

LCM of 9 and 12 = 2² × 3² = 4 × 9 = 36

HCF of 9 and 12 = 3

LCM (9 & 12) × HCF (9 & 12) = 36 × 3 = 108

Product of 9 and 12 = 9 × 12 = 108

Hence, LCM (9 & 12) × HCF (9 & 12) = 108 = 9 × 12

Example 2: 8 and 9 are two co-prime numbers. Using this numbers verify, LCM of Co-prime Numbers = Product Of The Numbers

Solution: LCM and HCF of 8 and 9:

8 = 2 × 2 × 2 = 2³

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

Example 3: Find the HCF of $\frac{12}{25}$, $\frac{9}{10}$, $\frac{18}{35}$, $\frac{21}{40}$

Solution: The required HCF is = $\frac{ HCF\: of\: 12, \: 9,\: 18, \: 21}{LCM \: of\: 25, \: 10,\: 35, \: 40}$ = $\frac{3}{1400}$