This problem requires finding the largest number $d$ such that dividing $55$, $103$, and $163$ by $d$ leaves the same remainder. Here’s how we can solve it:

Step 1: Restate the Problem Mathematically

If $d$ leaves the same remainder $r$ when dividing $55$, $103$, and $163$, then: $55 \mod d = 103 \mod d = 163 \mod d = r$ This implies: $(103 - 55) \mod d = 0 \quad \text{and} \quad (163 - 103) \mod d = 0 \quad \text{and} \quad (163 - 55) \mod d = 0$

Thus, $d$ must divide the differences: $103 - 55 = 48, \quad 163 - 103 = 60, \quad \text{and} \quad 163 - 55 = 108$

Step 2: Find the GCD of the Differences

We compute the greatest common divisor (GCD) of $48$, $60$, and $108$ to find the largest $d$.

Step 2.1: Compute GCD of $48$ and $60$

$\text{GCD}(48, 60) = 12$

Step 2.2: Compute GCD of $12$ and $108$

$\text{GCD}(12, 108) = 12$

Step 3: Verify the Result

The largest $d$ that divides $48$, $60$, and $108$ is $12$. Thus, $d = 12$ is the largest number such that dividing $55$, $103$, and $163$ by $d$ leaves the same remainder.

Final Answer:

The largest number is: $\boxed{12}$

Do you want a step-by-step breakdown of how the GCD was computed? Let me know if you have any further questions or need clarification!

Here are five related questions to expand your understanding:

1. How is the GCD of two numbers calculated using the Euclidean algorithm?
2. Can you find other divisors of $48$, $60$, and $108$ that satisfy the condition in the problem?
3. What happens if the remainders are not the same but have a fixed difference?
4. How does the concept of modular arithmetic relate to this problem?
5. Could this problem be solved using prime factorization instead of the GCD?

Tip: Practice computing the GCD using the Euclidean algorithm, as it’s a quick and efficient method for solving such problems!