Question

Find the HCF of 36, 54, and 90 using the prime factorization method.

Hint:

Steps for HCF using prime factors:
Write prime factorization of all numbers.
Pick common primes only.
Take smallest powers.
This method works for any number of values.

Answer 1

Concept: To find the HCF using prime factorization:
Express each number as a product of prime factors.
Take only the common prime factors with the smallest powers.
Step 1: Prime factorization 36: \[ 36 = 2^2 \times 3^2 \] 54: \[ 54 = 2 \times 27 = 2 \times 3^3 \] 90: \[ 90 = 9 \times 10 = 3^2 \times 2 \times 5 = 2 \times 3^2 \times 5 \] Step 2: Identify common prime factors Common primes in all three: \[ 2 \quad \text{and} \quad 3 \] Smallest powers: \[ 2^1, \quad 3^2 \] Step 3: Find HCF \[ \text{HCF} = 2 \times 3^2 = 2 \times 9 = 18 \] Final Answer: \[ \boxed{18} \]