Question

If HCF of 65 and 117 is expressed as 65p - 117, then the value of p will be

• A. 1
• B. 2
• C. 3
• D. 4

Hint:

For finding the HCF of two numbers, the Euclidean algorithm can be faster. Divide 117 by 65: `117 = 1 \times 65 + 52`. Then divide 65 by 52: `65 = 1 \times 52 + 13`. Then divide 52 by 13: `52 = 4 \times 13 + 0`. The last non-zero remainder, 13, is the HCF.

Answer 1

The Correct Option is B

Step 1: Understanding the Concept:
The question requires us to first find the Highest Common Factor (HCF) of 65 and 117. Then, we need to equate this HCF value to the given expression `65p - 117` to solve for the variable `p`.
Step 2: Key Formula or Approach:
We will use the prime factorization method to find the HCF of the two numbers.
HCF is the product of the lowest powers of common prime factors.
Step 3: Detailed Explanation:
First, find the prime factors of 65 and 117.
\[ 65 = 5 \times 13 \] \[ 117 = 3 \times 39 = 3 \times 3 \times 13 = 3^2 \times 13 \] The common prime factor between 65 and 117 is 13.
Therefore, the HCF of 65 and 117 is 13.
Now, according to the problem, the HCF is expressed as `65p - 117`.
So, we can set up the equation:
\[ 65p - 117 = 13 \] Add 117 to both sides of the equation:
\[ 65p = 13 + 117 \] \[ 65p = 130 \] Divide by 65 to find the value of `p`:
\[ p = \frac{130}{65} \] \[ p = 2 \] Step 4: Final Answer:
The value of `p` is 2.