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

#### Question 3: Find the LCM and HCF of the following integers by applying the prime factorisation method.

i. 12, 15 and 21

ii. 17, 23 and 29

iii. 8, 9 and 25

#### Explanatory Answer

##### i. 12, 15 and 21

Step 1: Prime factorize the numbers

12 = 2^{2} × 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) = 2^{2} × 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 = 2^{3}

9 = 3^{2}

25 = 5^{2}

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) = 2^{3} × 3^{2} × 5^{2} = 8 × 9 × 25 = 1800