The greatest number by which if 1657 and 2037 are divided the remainders will be 6 and 5 respectively, is-

Options

A. 127
B. 235
C. 260
D. 305

Answer

To find the greatest number by which 1657 and 2037 can be divided such that the remainders are 6 and 5 respectively, we can follow these steps:

### Step 1: Adjust the numbers by subtracting the remainders
We need to adjust both numbers by subtracting their respective remainders.

- For 1657, subtract 6:
\( 1657 - 6 = 1651 \)

- For 2037, subtract 5:
\( 2037 - 5 = 2032 \)

### Step 2: Find the HCF (Highest Common Factor) of the adjusted numbers
Now, we need to find the HCF of the adjusted numbers 1651 and 2032.

1. **Prime factorization of 1651**:
- Check divisibility by prime numbers:
- 1651 is not divisible by 2 (it's odd).
- Sum of digits (1+6+5+1=13) is not divisible by 3.
- It does not end in 0 or 5, so not divisible by 5.
- Check for 7: \( 1651 ÷ 7 \approx 235.857 \) (not divisible).
- Continue checking up to the square root of 1651 (approximately 40.6).
- 1651 is found to be a prime number.

2. **Prime factorization of 2032**:
- Check divisibility:
- 2032 is even, divide by 2: \( 2032 ÷ 2 = 1016 \)
- \( 1016 ÷ 2 = 508 \)
- \( 508 ÷ 2 = 254 \)
- \( 254 ÷ 2 = 127 \)
- 127 is a prime number.
- So, the prime factorization of 2032 is \( 2^4 \times 127 \).

### Step 3: Determine the HCF
Since 1651 is prime and does not share any factors with 2032 (which has the factors \( 2^4 \) and \( 127 \)), the only common factor is 127.

### Step 4: Conclusion
The greatest number by which 1657 and 2037 can be divided such that the remainders are 6 and 5 respectively is **127**.