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

 Aug 14, 2023
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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.

 Aug 14, 2023
edited by The3Mathketeers  Aug 14, 2023

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