H.C.F & L.C.M OF NUMBERS

Factors And Multiples :

If a number x divides another number y exactly, we say that x is factor of y. Also, in this case, y is called a multiple of x.

Highest Common Factor (H.C.F. Or G.C.D. or G.C.M) :

The H.C.F. of Two or more than Two numbers is the greatest number that divides each one of them exactly.
The Highest Common Factor is also known as Greatest Common divisor or Greatest Common Measure.

H.C.F. by Factorization :

Express each one of the given numbers as the product of prime factors. Now, choose common factors and take the product of these factors to obtain the required H.C.F.

For Example ;
Find the H.C.F. of 96, 528, 1584, and 2016.
Solution,96 = 2⁵ × 3 ; 528 = 2⁴ × 3 × 11 ; 1584 = 2⁴ × 3² × 11 ;
2016 = 2⁵ × 3² × 7
H.C.F. = 2⁴ × 3 = (16 × 3) = 48

H.C.F. By Division Method :

Suppose we have to find the H.C.F. of two given numbers. Divide the larger number by the smaller one. Now, divide the divisor by the remainder. Repeat the process of dividing the preceding divisor by the remainder last obtained, till a remainder zero is obtained. The last divisor is the H.C.F. of given numbers.

Suppose we have to find the H.C.F. of three numbers. Then H.C.F. of (H.C.F. of any two and the third number) gives the H.C.F. of given three numbers.

Similarly, The H.C.F. of more than three numbers may be obtained.

Lowest Common Multiple (L.C.M.) :

The least number which is exactly divisible by each one of the given numbers, is called their L.C.M.

FORMULA :

Product of Two Numbers = ( Their H.C.F.) × (Their L.C.M.).

L.C.M. By Factorization :

Resolve each one of the given numbers into prime factors. Then, L.C.M. is the product of highest powers of all the factors.

Note : L.C.M. of three numbers = L.C.M. of (L.C.M. of any two & third).

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

H.C.F. of Fractions = H.C.F. of Numerators / L.C.M. of Denominators.
L.C.M. of Fractions = L.C.M. of Numerators / H.C.F. of Denominators.