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

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Factors and Multiples:Suppose there are two numbers - a and b. If a number a divides another number exactly, we say that a is a factor of b and b is called a multiple of a.

Highest Common Factor (HCF) or Greatest Common Divisor (GCD)

The greatest common divisor (gcd), also known as the greatest common denominator, greatest common factor (gcf), or highest common factor (hcf), of two or more non-zero integers, is the largest positive integer that divides the numbers without a remainder. For example, the GCD of 8 and 12 is 4.

The HCF of two or more than two numbers is the greatest number that divides each of them exactly. There are two methods of finding the HCF of a given set of numbers:

• Factorization Method: In this method, express each one of the given numbers as the product of prime factors. The product of least powers of common prime factors gives HCF.

• Division Method: Divide the larger number by the smaller one. Now, divide the divisor by the remainder. Repeat the process of dividing the preceding number by the remainder last obtained till zero is obtained as remainder. The last divisor is the required HCF.

Finding the HCF of more than two numbers: H.C.F. of [(H.C.F. of any two) and (the third number)] gives the HCF of three given numbers.

Least Common Multiple (LCM):The lowest common multiple or (LCM) least common multiple or smallest common multiple of two rational numbers a and b is the smallest positive rational number that is an integer multiple of both a and b. The definition can be generalised for more than two numbers.

The least number which is exactly divisible by each one of the given numbers is called their LCM

• Factorization Method of Finding LCM: Resolve each one of the given numbers into a product of prime factors. Then, LCM is the product of highest powers of all the factors.

• Common Division Method (Short-cut Method) of Finding LCM: Arrange the given numbers in a row in any order. Divide by a number which divides exactly at least two of the given numbers and carry forward the numbers which are not divisible. Repeat the above process till no two of the numbers are divisible by the same number except 1. The product of the divisors and the undivided numbers is the required LCM of the given numbers.=

Product of two numbers = Product of their HCF and LCM 

Co-primes: Two numbers are said to be co-primes if their HCF is 1. HCF of two co-prime numbers is 1.

HCF and LCM of Fractions