What does "Solve $ax \equiv b \pmod{337}$ for $x$" mean? I have a general question about modular equations:
Let's say I have this simple equation:
$$ax\equiv b \pmod{337}$$
I need to solve the equation.
What does "solve the equation" mean? There are an infinite number of $x$'s that will be correct.
Does $x$ need to be integer?
Thanks very much in advance,
Yaron.
 A: Write  $p = 337$. This is a prime number.
First of all, if $a \equiv 0 \pmod{p}$ then there is a solution if and only if $b \equiv 0 \pmod{p}$, and then all integers $x$ are a solution.
If $a \not\equiv 0 \pmod{p}$, then the (infinite number of integer) solutions will form a congruence class modulo $p$. These can be found using Euclid's algorithm to find an inverse of $a$ modulo $p$, very much as you would do over the real numbers, say.
That is, use Euclid to find $u, v \in \Bbb{Z}$ such that $a u + p v = 1$, and then note that $x_{0} = u b$ is a solution, as $a x_{0} = a u b = b - p v b \equiv b \pmod{p}$. (Here $u$ is the inverse of $a$ modulo $p$, as $a u \equiv 1 \pmod{p}$.)
Then note that if $x$ is any solution, then $a (x - x_{0}) \equiv 0 \pmod{p}$, which happens if and only if $p \mid x - x_{0}$, so that $x \equiv x_{0} \pmod{p}$. Thus the set of solutions is the congruence class of $x_{0}$.
A: Typically this question means to find all the integer $x$ in the interval $[0,336]$ satisfying the equation.
