H.C.F. AND L.C.M. OF NUMBERS - Basic Concepts
IMPORTANT FACTS AND FORMULAE
I. Factors and Multiples :- If a number A(say) divides another number B(say) exactly, in that case we say that A is a factor of B and in this case, B is called a multiple of A.
II. Highest Common Factor (H.C.F.) or Greatest Common Measure (G.C.M.) or Greatest Common Divisor (G.C.D.): The H.C.F.(Highest Common Factor) of two or more than two numbers is the greatest number that divides each of them exactly.
There are mainly two methods for finding the H.C.F. of a given set of numbers and these are as follows:
1. Factorization Method : Express the each one of given numbers as the product of prime factors.The product of least powers of common prime factors gives the H.C.F.
2. 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 number by the remainder last obtained till zero is obtained as remainder.
- The last divisor is the required H.C.F.
Finding the H.C.F. of more than two numbers : In the case where we have to find the H.C.F. of three numbers, H.C.F. of [(H.C.F. of any two) and (the third number)] gives the H.C.F. of three given numbers and Similarly, the H.C.F. of more than three numbers may be obtained.
III. 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(Least Common Multiple).
There are mainly two methods for finding the L.C.M. of a given set of numbers and these are as follows:
1. Factorization Method of Finding L.C.M.:
- Resolve the each one of given numbers into a product of the prime factors.
- Then, the L.C.M. is the product of highest powers of all factors.
2. Common Division Method or Short-cut Method for Finding L.C.M.:
- Arrange given numbers in a row in any order.
- Divide by a number which divides exactly at least two of the given numbers & carry forward the numbers which are not divisible.
- Repeat above process till no two of the numbers are divisible by the same number except 1.
- The product of divisors and the undivided numbers is required L.C.M. of the given numbers.
IV. Product of two numbers =Product of their H.C.F. and L.C.M.
V. Co-primes: Two numbers are said to be co-primes if their H.C.F. is 1.
VI. H.C.F. and L.C.M. of Fractions:
- H C F= H.C.F. of Numerators / L.C.M. of Denominators
- L C M = L.C.M of Numerators / H.C.F. of Denominators
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Shubham Prajapati
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