Exercise 2.3.8 (First examples, part 8)

Question

Find the smallest positive integer giving remainders $1,2,3,4,$ and $5$ when divided by $3,5,7,9,$ and $11$, respectively.

Richard Ganaye · Accepted Answer

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

\begin{proof} We want to solve the system (S)
$$x \equiv 1 \pmod 3,\ x \equiv 2 \pmod 5,\ x \equiv 3 \pmod 7,\ x \equiv 4 \pmod 9,\ x \equiv 5 \pmod{11}.$$
Since $x\equiv 4 \pmod 9$ implies $x \equiv 1 \pmod 3$, we can forget the first congruence: (S) is equivalent to
$$ x \equiv 2 \pmod 5,\qquad x \equiv 3 \pmod 7,\qquad x \equiv 4 \pmod 9,\qquad x \equiv 5 \pmod{11}.
$$
where the moduli $5,7,9,11$ are relatively prime by pairs.

A solution is $x_0 = 1732$. So the solutions of (S) are
$$x = 1732 + k \, 3465,\ k\in \Z.$$
The smallest positive integer giving remainders $1,2,3,4,$ and $5$ when divided by $3,5,7,9,$ and $11$, respectively is $1732$.
\end{proof}

Exercise 2.3.8 (First examples, part 8)

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Exercise 2.3.8 (First examples, part 8)

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