congruence

It is wise to convert modular congruence to equations because we have more experience with equations and know how to manipulate them with ease. $3n \equiv 1356 \pmod 5 \iff \exists m \in \mathbb{Z} : 3n = 5m + 1356$.

$3n = 5m + 1356 \\ n = \frac{5m + 1356}{3}$

The problem seeks the smallest positive integer, so $n \geq 1$.

$n \geq 1 \\  \frac{5m + 1356}{3} \geq 1 \\  5m + 1356 \geq 3 \\  5m \geq -1353 \\ m \geq -\frac{1353}{5} = -270.6$

We know that $n = \frac{5m + 1356}{3} = \frac{5}{3}m + 452$. We must select the first value of m that is divisible by 3 so that n will be an integer. Since $m \geq -270.6$, the first possible value is m = -270. In this case,$n = \frac{5}{3} * -270 + 452 = -450 + 452 = 2$. In other words, $n = 2$ is the smallest positive integer n.