You have the linear Diophantine equation:

$177x + 52y = 1$

This is an equation where you need to find integer solutions for $x$ and $y$. The method to solve this is typically through the Extended Euclidean Algorithm, which can be used to find integer solutions to equations of the form $ax + by = \gcd(a, b)$.

Here are the steps:

Step 1: Apply the Euclidean algorithm to find the greatest common divisor (gcd) of 177 and 52.

1. Divide 177 by 52: $177 \div 52 = 3 \quad \text{(quotient)} \quad \text{remainder} = 177 - 3 \times 52 = 177 - 156 = 21$ So, $177 = 3 \times 52 + 21$.

2. Now, divide 52 by 21: $52 \div 21 = 2 \quad \text{(quotient)} \quad \text{remainder} = 52 - 2 \times 21 = 52 - 42 = 10$ So, $52 = 2 \times 21 + 10$.

3. Divide 21 by 10: $21 \div 10 = 2 \quad \text{(quotient)} \quad \text{remainder} = 21 - 2 \times 10 = 21 - 20 = 1$ So, $21 = 2 \times 10 + 1$.

4. Finally, divide 10 by 1: $10 \div 1 = 10 \quad \text{(quotient)} \quad \text{remainder} = 0$ So, $10 = 10 \times 1 + 0$.

Now that the remainder is 0, we stop. The gcd of 177 and 52 is 1, which is exactly what we want because 1 is the right-hand side of the equation.

Step 2: Use the Extended Euclidean Algorithm to express the gcd as a linear combination of 177 and 52.

We now backtrack through the Euclidean steps to write 1 as a combination of 177 and 52.

1. From $21 = 2 \times 10 + 1$, we can solve for 1: $1 = 21 - 2 \times 10$
2. Substitute $10 = 52 - 2 \times 21$ into the equation: $1 = 21 - 2 \times (52 - 2 \times 21)$ Simplify: $1 = 21 - 2 \times 52 + 4 \times 21 = 5 \times 21 - 2 \times 52$
3. Now, substitute $21 = 177 - 3 \times 52$ into this: $1 = 5 \times (177 - 3 \times 52) - 2 \times 52$ Simplify: $1 = 5 \times 177 - 15 \times 52 - 2 \times 52 = 5 \times 177 - 17 \times 52$

Thus, we have found that: $1 = 5 \times 177 - 17 \times 52$

Step 3: Solution

The integers $x = 5$ and $y = -17$ are a particular solution to the equation $177x + 52y = 1$.

General Solution

The general solution for the equation $177x + 52y = 1$ is given by: $x = 5 + 52t, \quad y = -17 - 177t$ for any integer $t$.

Would you like to see further details on how to interpret these solutions or explore specific cases?