Question

Find the HCF of 405, 585, 765 and 900.

• A. \(45 \)
• B. \(90 \)
• C. \(35 \)
• D. \(15 \)
• E. \(50 \)

Hint:

For HCF using prime factorization: - Take only the common prime factors - Use the smallest power of each common factor

Answer 1

The Correct Option is A

Concept: The Highest Common Factor (HCF) of numbers is the greatest number that divides all the given numbers exactly. It can be found using:
• Prime factorization method, or
• Repeated division (Euclidean algorithm)
Step 1: Prime factorization of each number.
\[ 405 = 3^4 \times 5 \] \[ 585 = 3^2 \times 5 \times 13 \] \[ 765 = 3^2 \times 5 \times 17 \] \[ 900 = 2^2 \times 3^2 \times 5^2 \]

Step 2: Identify common factors with the smallest powers.
Common prime factors in all numbers: \[ 3^2 \quad \text{and} \quad 5^1 \]

Step 3: Compute HCF.
\[ \text{HCF} = 3^2 \times 5 = 9 \times 5 = 45 \]