### Properties of HCF and LCM:

Some properties concerning the H.C.F. and the L.C.M. of numbers are:

**Property 1:** The H.C.F. of given numbers is not greater than any of the numbers.

**Verification:** H.C.F. of 161 and 345.

We can see that the H.C.F. of 161 and 345 is **23** which is **not greater than** any of the given numbers.

**Property 2:** The L.C.M. of given numbers is not less than any of the given numbers.

**Verification:** L.C.M. of 8 and 12.

L.C.M. of 161 and 345 is **24** which is **not less than** any of the given numbers.

**Property 3:** The H.C.F. of two co-prime numbers is **1.**

**Verification:** H.C.F. of 3 and 5.

We can see that the H.C.F. of 3 and 5 which are **co primes** is **1**.

**Property 4:** The L.C.M. of two or more co-prime numbers is equal to their product.

**Verification:** L.C.M. of 3 and 5.

L.C.M. of 3 and 5 is 3 x 5 x 1 = 15, which is the product of the co primes.

**Property 5:** If a number, say x, is a factor of another number, say y, then the H.C.F. of x and y is x and their L.C.M. is y.

**Property 6:** The H.C.F. of given numbers is always a factor of their LC.M.

**Property 7:** The product of the H.C.F. and the L.C.M. of two numbers is equal to the product of the given numbers. That is, if a and b are two numbers, then a x b = H.C.F. x L.C.M.

or, **L.C.M. = (a x b)/H.C.F., H.C.F. = (a x b)/L.C.M.**

**Verification:** L.C.M. and H.C.F. of 8 and 12.

L.C.M. = 2 x 2 x 2 x 3 = 24

H.C.F. = 2 X 2 = 4

L.C.M. X H.C.F. = 24 x 4 = 96

Product of given numbers = 8 x 12 = 96

Hence, product of the H.C.F. and the L.C.M. of two numbers is equal to the product of the given numbers.