Solve the following linear congruence: 17x congruence 3(mod 7)

Step 1
Consider the provided question,
We have to solve: ${\displaystyle 17x\equiv 3\left(\text{mod}7\right)}$
When divide 17 by 7, we get the remainder 3.
So, the above congruence is wtitten as,
${\displaystyle 3x\equiv 3\left(\text{mod}7\right)}$
here my aim is doing coefficient of x to be 1.
So, multiply both sides by 5.
${\displaystyle 15x\equiv 15\left(\text{mod}7\right)}$
Now, when divide 15 by 7, we get the remainder 1.
here in this case we divide in both the 15.
${\displaystyle 1x\equiv 1\left(\text{mod}7\right)}$
Thus, ${\displaystyle x\equiv 1\left(\text{mod}7\right)}$
Step 2
The above solution is also written as,
The solution of ${\displaystyle 17x\equiv 3\left(\text{mod}7\right)}$ is,
${\displaystyle x\equiv 1\left(\text{mod}7\right)}$
It can be also written as,
x=1+7k where, k = 0,1,2,3,...