# L.C.M. and H.C.F.

### Prime factorisation

If a natural number is expressed as the product of prime numbers, then the factorisation of the number is called its*prime (or complete) factorisation.*

A prime factorisation of a natural number can be expressed in the

**exponential form.**

For example:

(i) 48 = 2×2×2×2×3 = 2

^{4}×3

(ii) 420 = 2×2×3×5×7 = 2² ×3×5×7.

**Least Common Multiple**(abbreviated

**L.C.M.**) of two natural numbers is the smallest natural number which is a multiple of both the numbers.

**Highest Common Factor**(abbreviated

**H.C.F.**) of two natural numbers is the largest common factor (or divisor) of the given natural numbers. In other words, H.C.F. is the greatest element of the set of common factors of the given numbers.

H.C.F. is also called

**Greatest Common Divisor**(abbreviated

**G.C.D.**)

**Co-prime numbers:**Two natural numbers are called

**co-prime numbers**if they have no common factor other than 1.

*In other words, two natural numbers are co-prime if their H.C.F. is*1.

Some examples of co-prime numbers are: 4, 9; 8, 21; 27, 50.

**Relation between L.C.M. and H.C.F. of two natural numbers**

*The product of L.C.M. and H.C.F. of two natural numbers = the product of the numbers*.

**Note.**

In particular, if two natural numbers are co-prime then their L.C.M. = the product of the numbers.

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