To find the inverse of $7 \mod 25$, we need to determine an integer $x$ such that:

$7x \equiv 1 \pmod{25}$

This is the same as solving the linear Diophantine equation:

$7x - 25k = 1$

for integers $x$ and $k$. This can be done using the Extended Euclidean Algorithm.

Step 1: Apply the Euclidean Algorithm

We will find the greatest common divisor (GCD) of 7 and 25 using the Euclidean algorithm:

$25 = 3 \times 7 + 4$ $7 = 1 \times 4 + 3$ $4 = 1 \times 3 + 1$ $3 = 3 \times 1 + 0$

Since the GCD is 1, 7 and 25 are coprime, meaning an inverse exists.

Step 2: Apply the Extended Euclidean Algorithm

Now, we will work backwards to express 1 as a linear combination of 7 and 25.

$1 = 4 - 1 \times 3$ Substitute $3 = 7 - 1 \times 4$: $1 = 4 - 1 \times (7 - 1 \times 4) = 2 \times 4 - 1 \times 7$ Substitute $4 = 25 - 3 \times 7$: $1 = 2 \times (25 - 3 \times 7) - 1 \times 7 = 2 \times 25 - 7 \times 7$

Thus, we have: $1 = 2 \times 25 - 7 \times 7$

Step 3: Conclusion

From this, we see that the coefficient of $7$ is $-7$. Thus, the inverse of 7 modulo 25 is:

$-7 \equiv 18 \pmod{25}$

Therefore, the inverse of $7 \mod 25$ is $\boxed{18}$.

Would you like further details on the Extended Euclidean Algorithm or any other steps? Here are some related questions:

1. How do you verify that 18 is the correct inverse modulo 25?
2. What is the significance of the greatest common divisor (GCD) in modular arithmetic?
3. How would you compute an inverse for a larger modulus?
4. Can the Extended Euclidean Algorithm solve any Diophantine equation?
5. How do we know if an inverse does not exist in modular arithmetic?

Tip: An inverse modulo exists if and only if the number and the modulus are coprime (i.e., their GCD is 1).