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Factors and Multiples :

If a number a divides another number b exactly, we say that a is a factor of b. In this case, b is called a multiple of a.

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 divided each of them exactly.

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

• Factorization Method :

  Express the each one of the given numbers as the product of prime factors. The product of least powers of common prime factors gives H.C.F.

• Division Method :

  Suppose we have to find the H.C.F. of two given numbers, divide the larger 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 required H.C.F.

Least 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.

There are two methods of finding the L.C.M. of a given set of numbers :

• Factorization Method :

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

• Division Method (short-cut) :

  Arrange the given numbers in a rwo in any order. Divide by a number which divided 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 L.C.M. of the given numbers.

Product of two numbers = Product of their H.C.F. and L.C.M.

Co-primes :

Two numbers are said to be co-primes if their H.C.F. is 1.

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

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

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