Question

If H.C.F. of (65, 117) is expressed in the form \( 65m + 117n \), then the value of \( m \) is:

• A. 1
• B. 2
• C. 3
• D. 4

Hint:

Use the Extended Euclidean Algorithm to express HCF as a linear combination of given numbers.

Answer 1

The Correct Option is B

We find the HCF(65, 117) using the Euclidean Algorithm: \[ 117 \div 65 = 1, \quad \text{remainder} = 52 \] \[ 65 \div 52 = 1, \quad \text{remainder} = 13 \] \[ 52 \div 13 = 4, \quad \text{remainder} = 0 \]
Thus, HCF(65, 117) = 13. Now, express 13 as a linear combination: \[ 13 = 65 - 52 \times 1 \] Since 52 = 117 - 65 \times 1, substitute: \[ 13 = 65 - (117 - 65 \times 1) \times 1 \] \[ 13 = 65 - 117 + 65 \] \[ 13 = 2 \times 65 - 117 \]
Thus, comparing with \( 65m + 117n = 13 \), we get: \[ m = 2 \]
Thus, the correct answer is 2.