Prime Factorisation, HCF and LCM

Prime factorisation:
Writing a number as a product of its prime factors is called the prime factorisation of the number.
e.g. (i)  Prime factorisation of 18 = 2 x 3 x 3
      (ii)  Prime factorisation of 40 = 2 x 2 x 2 x 5

Highest common factor:
The greatest of the common factors of the given numbers is called their highest common factor (HCF). It is also known as the greatest common divisor. The HCF of the given numbers is equal to the product of common factors in their prime factorisation.
e.g.
Factors of 16 = 1, 2, 4, 8, 16
Factors of 40 = 1, 2, 4, 5, 8, 10, 20, 40
Common factors = 1, 2, 4, 8
HCF of 16 and 40 = 8

The HCF of 16 and 40 using their prime factorisation:
Prime factorisation of 16 = 2 x 2 x 2 x 2
Prime factorisation of 40 = 2 x 2 x 2 x 5
Common prime factors = 2 x 2 x 2
HCF of 16 and 40 = 2 x 2 x 2 = 8

Least common multiple:
The smallest common multiple of the given numbers is called their least common multiple (LCM). The product of the prime factors that occur the maximum number of times in the prime factorisation of the given numbers, is their LCM.
e.g.
The LCM of given numbers using their prime factorisation:
Prime factorisation of 4 = 2 x 2
Prime factorisation of 6 = 2 x 3
LCM of 4 and 6 = 2 x 2 x 3 =12

To find the LCM of the given numbers using the division method:

• Write the given numbers in a row.
• Divide the numbers by the smallest prime number that divides one or more of the given numbers.
• Write the number that is not divisible, in the second row.
• Write the new dividends in the second row.
• Divide the new dividends by another smallest prime number.
• Continue dividing till the dividends are all prime numbers or 1.
• Stop the process when all the new dividends are prime numbers or 1.
• The product of all the divisors and the remaining prime dividends is the LCM of the given numbers.