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.