Exercise 2.3.7 (First examples, part 7)

Question

Determine whether the congruences $5x \equiv 1 \pmod 6$, $4x \equiv 13 \pmod {15}$ have a common solution, and find them if they exist.

Richard Ganaye · Accepted Answer

\begin{proof} 
$$
\left\{
\begin{array}{l}
5x \equiv 1 \pmod 6\
4x \equiv 13 \pmod {15} 
\end{array}
ight.
\iff 
\left\{
\begin{array}{l}
x \equiv -1 \pmod 6\
x \equiv 7 \pmod {15} 
\end{array}
ight.
$$
The first congruence implies $x \equiv -1 \pmod 3$, and the second implies $x \equiv 1 \pmod 3$. Since $1
ot \equiv -1 \pmod 3$, this is a contradiction, so the system is inconsistent.
\end{proof}

Exercise 2.3.7 (First examples, part 7)

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Exercise 2.3.7 (First examples, part 7)

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