## Friday, 26 May 2017

### LCM AND HCF PROBLEMS EXPLANATION

For better understanding of the concept LCM (Lowest Common Multiple) and HCF (Highest Common Factor) we need to recollect terms multiples and factors. Let’s learn about LCM, HCF and relation between HCF and LCM of natural numbers.
Multiples: A multiple is any number which is exactly divisible by a given number. Ex: 3, 6,9,12, etc are the multiples of 3.
Factors:  A factor is a number which divides any given number without leaving remainder. Ex: 2,3,4,6,8,12 are the factors of 24.
Lowest Common Multiple (LCM):  The least or smallest common multiple of any two or more given natural numbers are termed as LCM. For example, LCM of 10, 15, and 20 is 60.
Highest Common Factor (HCF): The largest or greatest factor common to any two or more given natural numbers is termed as HCF of given numbers. Also known as GCD (Greatest Common Divisor).For example, HCF of 4, 6 and 8 is 2.
$4=&space;2\times&space;2$
$6=&space;3\times&space;2$
$8=&space;4\times&space;2$
Here, highest common factor of 4, 6 and 8 is 2.
Both HCF and LCM of given numbers can be find by using two methods; they are division method and prime factorization.
##### PROPERTIES OF HCF AND LCM
Property 1: The product of LCM and HCF of any two given natural numbers is equivalent to the product of the given numbers.
LCM x HCF = Product of the Numbers
Suppose A and B are two numbers, then.
LCM (A&B) x HCF (A&B) = A x B
Example 1: Prove that:
LCM (9 & 12) x HCF (9&12) = Product of 9 and 12
Solution: LCM and HCF of 9 and 12:
Property 2: HCF of co-prime numbers is 1. Therefore LCM of given co-prime numbers is equal to the product of the numbers.
LCM of Co-prime Numbers = Product Of The Numbers
Example 2: 8 and 9 are two co-prime numbers. Using this numbers verify,
LCM of Co-prime Numbers = Product Of The Numbers
Solution: LCM and HCF of 8 and 9:
Property 3: H.C.F. and L.C.M. of Fractions

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