LCM & HCF of 3 numbers - Prime Factorization

NCERT Solutions for Class 10 Maths | Chapter: Real Numbers | Concept: Compute LCM & HCF by Prime Factorization. Computing LCM and HCF for 3 positive integers

Explanatory Answer

i. 12, 15 and 21

Step 1: Prime factorize the numbers
12 = 2² × 3
15 = 3 × 5
21 = 3 × 7

Step 2: Compute HCF: HCF = Product of the lowest powers of common primes in the 3 numbers.
HCF (12, 15, 21) = 3

Step 3: Compute LCM: LCM = Product of the highest powers of all the primes in the 3 numbers.
LCM (12, 15, 21) = 2² × 3 × 5 × 7 = 420

ii. 17, 23 and 29

Step 1: Prime factorize the numbers
All 3 are prime numbers. So, there is nothing to be done in step 1.

Step 2: Compute HCF: HCF = Product of the lowest powers of common primes in the 3 numbers.
∴ HCF (17, 23, 29) = 1
Note: If all the given numbers are prime, they will have no common factor other than 1. So, HCF will be 1 for two or more prime numbers.

Step 3: Compute LCM: LCM = Product of the highest powers of all the primes in the 3 numbers.
LCM (17, 23, 29) = 17 × 23 × 29 = 11339
Note: If the given numbers are all prime, their LCM will be the product of the numbers.

iii. 8, 9 and 25

Step 1: Prime factorize the numbers
8 = 2³
9 = 3²
25 = 5²

Step 2: Compute HCF: HCF = Product of the lowest powers of common primes in the 3 numbers.
No prime factor common to the 3 numbers.
∴ HCF (8, 9, 25) = 1

Step 3: Compute LCM: LCM = Product of the highest powers of all the primes in the 3 numbers.
LCM (8, 9, 25) = 2³ × 3² × 5² = 8 × 9 × 25 = 1800