Finding ALL solutions of the modular arithmetic equation $25x \equiv 10 \pmod{40}$ I am unsure how to solve the following problem. I was able to find similar questions, but had trouble understanding them since they did not show full solutions.
The question:
Find ALL solutions (between $1$ & $40$) to the equation $25x \equiv 10 \pmod{40}$.

Let's use the definition of congruence.  $a \equiv b \pmod{n} \iff a = b + kn$ for some integer $k$.  Hence, $25x \equiv 10 \pmod{40}$ means $$25x = 10 + 40k$$ for some integer $k$.  Dividing each side of the equation $25x = 10 + 40k$ by $5$ yields $$5x = 2 + 8k$$ 
for some integer $k$.  Thus, 
$$5x \equiv 2 \pmod{8}$$
Since $\gcd(5, 8) = 1$, $5$ has a multiplicative inverse modulo $8$.  To isolate $x$, we must multiply both sides of the congruence $5x \equiv 2 \pmod{8}$ by the multiplicative inverse of $5$ modulo $8$.  To find the multiplicative inverse, we use the extended Eucldean algorithm.
\begin{align*}
8 & = 5 + 3\\
5 & = 3 + 2\\
3 & = 2 + 1\\
2 & = 2 \cdot 1 
\end{align*}
Working backwards through this partial sequence of Fibonacci numbers to solve for $1$ as a linear combination of $5$ and $8$ yields
\begin{align*}
1 & = 3 - 2\\
  & = 3 - (5 - 3)\\
  & = 2 \cdot 3 - 5\\
  & = 2(8 - 5) - 5\\
  & = 2 \cdot 8 - 3 \cdot 5
\end{align*}
Therefore, $1 \equiv -3 \cdot 5 \pmod{8}$.  Hence, $-3 \equiv 5^{-1} \pmod{8}$.  Since $-3 \equiv 5 \pmod{8}$, we have $5 \equiv 5^{-1} \pmod{8}$.  Thus, $5 \cdot 5x \equiv x \pmod{8}$.  Hence,
\begin{align*}
5x & \equiv 2 \pmod{8}\\
5 \cdot 5x & \equiv 5 \cdot 2 \pmod{8}\\
x & \equiv 10 \pmod{8}\\
x & \equiv 2 \pmod{8}
\end{align*}
What remains is for you to find the solutions of the congruence $x \equiv 2 \pmod{8}$ such that $0 \leq x < 40$.   
