Question

What is the HCF of 1241 and 1314.

• A. 37
• B. 73
• C. 43
• D. 83

Hint:

A useful property of HCF is that the HCF of two numbers must also divide their difference.
Here, the difference is \( 1314 - 1241 = 73 \).
Since 73 is a prime number, the HCF must be either 1 or 73. Testing divisibility by 73 directly yields the correct answer.

Answer 1

The Correct Option is B

Step 1: Understanding the Concept:
The Highest Common Factor (HCF) of two integers is the largest positive integer that divides both numbers without leaving a remainder.
A highly efficient method for finding the HCF of two large numbers is the Euclidean Division Algorithm.
Key Formula or Approach:
The Euclidean Algorithm states that for any two positive integers \( a \) and \( b \) (where \( a > b \)):
\[ a = b \cdot q + r, \quad \text{where } 0 \le r < b \] Then, \( \text{HCF}(a, b) = \text{HCF}(b, r) \).
This process is repeated until the remainder becomes zero. The non-zero remainder in the last step is the HCF.

Step 2: Detailed Explanation:
Let us apply the algorithm to the given numbers \( a = 1314 \) and \( b = 1241 \):
1. Divide 1314 by 1241:
\[ 1314 = 1241 \times 1 + 73 \] Here, the quotient is 1 and the remainder is 73.
2. Now, take the divisor 1241 and the remainder 73, and divide 1241 by 73:
\[ 1241 = 73 \times 17 + 0 \] Let us double-check the multiplication:
\[ 73 \times 17 = 73 \times (10 + 7) = 730 + 511 = 1241 \] Since the remainder has become 0, the divisor at this stage is the HCF.
Therefore, the HCF of 1241 and 1314 is 73.

Step 3: Final Answer:
The correct option is (B).