HCF and LCM

HCF (Highest Common Factor) : 

HCF of two numbers is the largest common factor of both the numbers. 

Example :

Factors of 12 : 1, 2, 3, 4, 6, 12

Factors of 15 : 1, 3, 5, 15

12 and 15 have 1 and 3 as common factors. Among these common factors 3 is the highest. Hence, the HCF of 12 and 15 is 3

How to Find HCF?

HCF can be found using the prime factorisation method or by using the division method.

Prime Factorisation Method :

  1. Express the numbers in terms of their prime factors
  2. Take the product of the least power of the common prime factors, which gives you the HCF

Division Method :

HCF of 2 numbers using division method can be obtained as

  1. Divide the larger of the two numbers by smaller number
  2. Divide the divisor by the remainder obtained in step 1
  3. Repeat the steps until remainder obtained is 0.
  4. The last divisor which gives the remainder 0 is the HCF.

If you have to find HCF of more than 2 numbers, say 3 numbers.

  1. First find HCF of any two numbers. Let the HCF obtained be H.
  2. Find the HCF of H and the third number.
  3. Value obtained in step 2 is the HCF of the three given numbers.

Apply the same strategy to numbers greater than 3.

LCM or Least Common Multiple :

LCM of two numbers is the smallest number which is the multiple of both the numbers.

Example:

Multiples of 2 : 2, 4, 6, 8, 10, 12, 14, 18 …

Multiples of 3 : 3, 6, 9, 12, 15, 18…

2 and 3 have 6, 12, 18 … as common multiples. Among these common multiples 6 is the least common multiple. Hence LCM of 2 and 3 is 6.

How to find LCM

HCF can be found using the prime factorisation method or by using the division method.

Prime Factorisation Method :

  1. Express the numbers in terms of their prime factors
  2. Take the product of the highest power of the common prime factors, which gives you the LCM

Division Method :

  1. Write all the numbers in a row in any order
  2. Divide the numbers by a number which divides any two of the numbers. All the other numbers are written as it is. The values obtained will be written below the corresponding numbers written in step 1.
  3. Repeat the steps 1 and 2 until no two numbers in a row are divisible by same number other than 1.
  4. Take the product of the divisors and the undivided numbers in the last row, which gives you the LCM

Relation between two numbers and their HCF and LCM

If a and b are two numbers, then

a x b = HCF(a,b) x LCM(a,b)

Co-prime Numbers

Two numbers are said to be co-prime if the HCF of both the numbers is 1.

Example : Consider 5 and 12. Their HCF is 1. Hence they are co-primes

HCF and LCM of Fractions:

  1. HCF = $\frac{\text{HCF of Numerators}}{\text{LCM of Denominators}}$
  2. LCM = $\frac{\text{LCM of Numerators}}{\text{HCF of Denominators}}$