+817 What is the smallest positive integer $n$ such that $3n \equiv 1356 \pmod{5}?$
+177 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.
