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

Factor: If a number ‘A’ divides another number ‘B’ exactly, we say that A is a factor of B.

Example: 8 and 9 are factors of 72.

Multiple: A number is said to be a multiple of another, when it is exactly divisible by the other.

Example: 72 is multiple of 8 and 9.

Prime Numbers:

A counting number is called a prime number if it has exactly two factors namely itself and 1. 

Ex: All prime numbers below 100 are 

2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97

Composite Numbers:

The natural numbers which are not prime numbers, are called Composite Numbers. 

Ex: 4, 6, 8, 9, 10, .......

Co-Primes:

Two natural numbers 'a' and 'b' are said to be C0-Primes (Prime-to-Each other) if their HCF is '1' (or) they have no common factors except unity.

 Ex: (2,1),  (2,3),  (3,5),  (7,9),  (5,11), .......

Highest Common Factor (H.C.F) or Greatest Common Measure (G.C.M) or Greatest Common Divisor (G.C.D):

The H.C.F of two or more than two numbers is the greatest number that divides each of them exactly. 

There are two methods of finding H.C.F of a given set of numbers:

Methods of finding H.C.F:

    1. HCF by Factorization method:

 Express each of the given number as the product of Prime Factors.

 Choose the common Factors of those numbers.

 Find the product of least powers of the common Prime Factors.

 This product gives the H.C.F of the given numbers.

Ex: Find the H.C.F of 108, 288, 360

108 --> 2 X 2 X 3 X 3 X 3 = 2² X 3³

288 --> 2 X 2 X 2 X 2 X 2 X 3 X 3 = 2⁵ X 3²

360 --> 2 X 2 X 2 X 5 X 3 X 3 = 2³ X 5 X 3²

From the above, Common Factors of 108, 288, 360 are 2, 3.

Now, the product of Lowest powers of common factors 2² X 3² = 36.

2. HCF by Successive Division method:

ØTake two different numbers.

ØDivide the larger number by smaller number.

ØNow, divide the divisor by the remainder.

ØRepeat this process of dividing the preceding divisor by the remainder last obtained till you get the remainder “zero”.

ØThe last divisor is the HCF of the given numbers. 

Ex: Find the H.C.F of 108, 288.

Least Common Multiple (L.C.M) :

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

--> Product of two numbers (First number x Second Number) = Product of their  H.C.F and L.C.M      (i.e., H.C.F. X L.C.M).

--> H.C.F. of a given number always divides its L.C.M.

--> Largest number which divides x, y, z to leave remainder R in each case = H.C.F. of (x-R), (y-R), (z-R).

--> Largest number which divides x, y, z to leave same remainder = H.C.F. of (y-x), (z-y), (z-x).

--> Largest number which divides x, y, z to leave remainder a, b, c = H.C.F. of (x-a), (y-b), (z-c).

--> Least number which when divided by x, y, z and leaves a remainder R in each case = (L.C.M. of x, y, z) + R