Find the largest number which when it divides 1545 and 2091, leaves a remainder of 9 and 11 respectively.

Answer

To find the largest number that divides 1545 and 2091, leaving a remainder of 9 and 11 respectively, we can follow these steps:

### Step-by-Step Solution:

1. **Understand the Problem**: We need to find a number \( x \) such that:
- \( 1545 \mod x = 9 \)
- \( 2091 \mod x = 11 \)

2. **Subtract the Remainders**: Since the number leaves remainders, we can subtract these remainders from the original numbers:
- For 1545: \( 1545 - 9 = 1536 \)
- For 2091: \( 2091 - 11 = 2080 \)

3. **Identify the Numbers to Find GCD**: Now we need to find the greatest common divisor (GCD) of the two results:
- We need to find \( \text{GCD}(1536, 2080) \).

4. **Use the Euclidean Algorithm**: We will apply the Euclidean algorithm to find the GCD:
- Start with \( 2080 \) and \( 1536 \):
- \( 2080 = 1536 \times 1 + 544 \)
- Now take \( 1536 \) and \( 544 \):
- \( 1536 = 544 \times 2 + 448 \)
- Next, take \( 544 \) and \( 448 \):
- \( 544 = 448 \times 1 + 96 \)
- Now take \( 448 \) and \( 96 \):
- \( 448 = 96 \times 4 + 64 \)
- Next, take \( 96 \) and \( 64 \):
- \( 96 = 64 \times 1 + 32 \)
- Finally, take \( 64 \) and \( 32 \):
- \( 64 = 32 \times 2 + 0 \)

5. **Conclusion**: The last non-zero remainder is \( 32 \). Therefore, the GCD of \( 1536 \) and \( 2080 \) is \( 32 \).

6. **Final Answer**: The largest number which divides \( 1545 \) and \( 2091 \), leaving remainders \( 9 \) and \( 11 \) respectively, is \( 32 \).