Exercise 2.3.5 (First examples, part 5)

Question

Solve Example 2 using the technique that was applied to Example 3, second solution (*).

(*) There is a misprint in the sentence(R.G.).

Richard Ganaye · Accepted Answer

ewcommand{\Z}{\mathbb{Z}}

\begin{proof} The system to solve is
\begin{align}x \equiv 29 \pmod{52},\qquad x \equiv 19 \pmod {72},
\end{align}
which is inconsistent, by example 2.

Following the technique  that was applied to Example 3, second solution, the second equation is equivalent to
$$x = 19 + u 72,\ u \in \Z.$$
On substituting this into the first congruence, we obtain
$$19 + u 72 \equiv 29 \pmod{52},$$
that is
$$  72 u\equiv 10 \pmod{52},$$
Since $\mathrm{gcd}(72,52) = 4 
mid 10$, there is no solution to system (1).
\end{proof}

Exercise 2.3.5 (First examples, part 5)

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Exercise 2.3.5 (First examples, part 5)

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