Q3) Find the least common multiple (LCM) and highest common factor (HCF) by prime factorization method:
(i) 60, 76, 106

To find the Least Common Multiple (LCM) and the Highest Common Factor (HCF) of the numbers 60, 76, and 106 using prime factorization, we'll break each number down into its prime factors and use this information to calculate the results.
Step 1: Prime Factorization

60 :

Divide by 2: 60 ÷ 2 = 30
Divide by 2: 30 ÷ 2 = 15
Divide by 3: 15 ÷ 3 = 5
Divide by 5: 5 ÷ 5 = 1

So, the prime factorization of 60 is: 2 2 × 3 1 × 5 1 .

76 :

Divide by 2: 76 ÷ 2 = 38
Divide by 2: 38 ÷ 2 = 19
Since 19 is prime, we have reached the end.

So, the prime factorization of 76 is: 2 2 × 1 9 1 .

106 :

Divide by 2: 106 ÷ 2 = 53
Since 53 is prime, we have reached the end.

So, the prime factorization of 106 is: 2 1 × 5 3 1 .


Step 2: Calculating LCM and HCF
LCM (Least Common Multiple):

Take the highest power of all prime numbers present in the factorizations.
LCM = 2 2 × 3 1 × 5 1 × 1 9 1 × 5 3 1

L CM = 4 × 3 × 5 × 19 × 53
L CM = 30180



HCF (Highest Common Factor):

Take the lowest power of all common prime numbers in all factorizations.
The only prime number common to all is 2. The lowest power of 2 is 2 1 .
HCF = 2


So, the LCM of 60, 76, and 106 is 30180, and the HCF is 2.

Answered by EmmaGraceJohnson | 2025-07-06