# L.C.M and H.C.F.

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## Important points for L.C.M and H.C.F.

#### L.C.M. (Least Common Multiple)

L.C.M. of two or more numbers is the smallest number which is exactly divisible by each of the numbers.

Methods to find L.C.M.:
1) Factorization method: Express each of the numbers as the product of prime numbers. L.C.M. is the product of highest powers of the prime numbers.

2) Division method
Example: L.C.M. of 2x2y3z7 and 3xy4z3 is 6x2y4z7 where x,y,z are the prime factors.

#### H.C.F. (Highest Common Factor) or G.C.D. (Greatest Common Divisor)

H.C.F. of two or more numbers is the greatest number which is present in each of the numbers.

Methods to find H.C.F.:
1) Factorization method to find H.C.F.: Express each of the numbers as the product of prime numbers. H.C.F. is the product of least powers of the common prime numbers.

2) Division method is the easiest method to find the H.C.F. of two numbers 'a' and 'b' is to divide the larger number by the smaller number. 'Divisor' of the first step becomes the 'dividend' of the second step and 'remainder' of the first step becomes the 'divisor' of the second step and soon. This process is carried on till the time remainder becomes 'zero'. Divisor of the step in which the remainder comes out to be zero is the H.C.F. of the numbers 'a' and 'b'.

3) Difference method To find the HCF of the given numbers you can divide the numbers by their lowest possible difference. If these numbers are divisible by this difference, then this difference itself is the HCF of the given numbers otherwise any other factor of this difference will be its HCF.

Example: Find the HCF of 30, 42 and 135.
We can notice that the difference between 30 and 42 is less than difference between 135 and 42 or 30. Difference between 42 and 30 is 12, but 12 does not divide 30, 42 and 135 completely.
Factors of 12 are 12, 6, 4, 3, 2, 1.
Clearly, 3 is the highest factor, which divides all the numbers completely.
Therefore, 3 is the HCF of 30, 42 and 135.
Example: H.C.F. of 2x2y3z7 and 3xy4z3 is xy3z3 where x,y,z are the prime factors.

#### Finding H.C.F. of more than two numbers:

Suppose there are three numbers a, b, c
Find the H.C.F. of a, b. Suppose H.C.F.(a, b) = h1
Now find H.C.F. of (h1, c). Suppose H.C.F.(h1, c) = h2
So, we can say that the H.C.F.(a, b, c) = h2

#### Finding L.C.M. and H.C.F. of fractions:

L.C.M. of fractions H.C.F. of fractions NOTE: 1) All the fractions of 2 or more must be in their lowest terms. If they are not in their lowest terms, then conversion in the lowest form is required before finding the HCF or LCM.
2) The required HCF of 2 or more fractions is the highest fraction which exactly divides each of the fractions. H.C.F of coprime numbers is 1.
3) The required LCM of 2 or more fractions is the least fraction/integer which is exactly divisible by each of them.
4) The HCF of numbers of fractions is always a fraction but this is not true in case of LCM.

H.C.F. × L.C.M. = Product of numbers. (This formula is applicable only for 2 numbers.)