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hcf--lcm-1

HCF (==Highest Common Factor==):

Also known as GCD (==Greatest Common Divisor==), it is the largest positive integer that divides two or more integers without leaving a remainder.

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How to Find HCF?

  1. HCF by prime factorization

  2. HCF by division method

  3. HCF by prime factorization

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  1. HCF by division method

Two numbers

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Three Numbers

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Euclidean Algorithm

As per the Euclidean Algorithm,

HCF(X, Y) = HCF(Y, X mod Y)

where X > Y and mod is the modulo operator.

HCF of Prime Numbers

We know that a prime number has only two factors, 1 and the number itself.

Let us consider two prime numbers 2 and 7, and find their HCF by listing their factors.

The factors of 2 = 1, 2; and

the factors of 7 = 1, 7.

We can see that the only common factor of 2 and 7 is 1.

Hence, the HCF of prime numbers is always equal to 1.

Properties of HCF

  • The HCF of two or more numbers divides each of the numbers without a remainder.
  • The HCF of two or more numbers is a factor of each of the numbers.
  • The HCF of two or more numbers is always less than or equal to each of the numbers.
  • The HCF of two or more prime numbers is 1 always.

Relation Between LCM and HCF

LCM (a,b) × HCF (a,b) = a × b

LCM (Least Common Multiple):

It is the smallest positive integer that is divisible by two or more integers

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How to Find LCM?

  • LCM by Prime Factorization Method
  • LCM using Division Method

  1. LCM by Prime Factorization Method

  2. LCM using Division Method

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Properties

  1. The HCF of any given numbers is never greater than any of the numbers
  2. The LCM of given numbers is not less than any of the given numbers.
  3. The HCF of co-prime numbers is always 1
    1. Two numbers are said to be Co Primes if their HCF is 1
  4. LCM of given co-prime numbers is always equal to the product of the numbers
  5. The product of LCM and HCF of any two given natural numbers is always equal to the product of those given numbers.
  6. H.C.F. and L.C.M. of Fractions

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