Question

The largest number which divides 245 and 1037, leaving remainder 5 in each case, will be:

• A. 22
• B. 23
• C. 24
• D. 25

Hint:

When dividing numbers leaving a specific remainder, subtract the remainder from both numbers and find the HCF of the new values.

Answer 1

The Correct Option is C

We are asked to find the largest number that divides both 245 and 1037, leaving a remainder of 5. This can be written as: \[ 245 = q_1 \cdot x + 5 \] \[ 1037 = q_2 \cdot x + 5 \] Subtracting these two equations: \[ 1037 - 245 = (q_2 - q_1) \cdot x \] \[ 792 = (q_2 - q_1) \cdot x \] Now, we need to find the greatest divisor of 792. The factors of 792 are: \[ 1, 2, 3, 4, 6, 8, 9, 11, 12, 16, 18, 22, 24, 33, 36, 44, 48, 66, 72, 88, 99, 132, 198, 264, 396, 792 \] We check the options: - \( 792 \div 24 = 33 \), so 24 divides 792. Thus, the largest divisor is 24, and the answer is \( \boxed{24} \).