A linear congruence

#linear congruence#inverse#solution

Statement

Solve $3x\equiv4\pmod7$ .

Hint

Find $3^{-1}\pmod7$ or test residues.

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Inverse. $3\cdot5=15\equiv1\pmod7$ , so $3^{-1}\equiv5$ .

Solution. $x\equiv5\cdot4=20\equiv6\pmod7$ .

Conclusion. $x\equiv6\pmod7$ , i.e. $x=6+7k$ , $k\in\mathbb{Z}$ .

Check. $3\cdot6=18\equiv4\pmod7$ . Correct.