Exercise 2.1.15 ($x$ satisfies at least one of the given congruences)

Question

Find integers $a_1,\ldots,a_5$ such that every integer $x$ satisfies at least one of the congruences $x \equiv a_1 \pmod 2, x \equiv a_2 \pmod 3, x \equiv a_3 \pmod 4, x \equiv a_4 \pmod 6, x \equiv a_5 \pmod{12}$.

Richard Ganaye · Accepted Answer

\begin{proof} We want to exhaust all classes modulo $12$.

Take $(a_1,a_2,a_3,a_4,a_5) = (1,2,0,4,6)$.

Then 
\begin{align*}
x \equiv 1 \pmod{2} & \iff x \equiv 1,3,5,7,9,11 \pmod{12},\
x \equiv 2 \pmod 3 & \iff x \equiv  2,5,8,11 \pmod {12},\
x \equiv 0 \pmod 4 & \iff x \equiv 0 , 4, 8 \pmod{12},\
x \equiv 4 \pmod 6 &\iff x \equiv 4, 10 \pmod{12,}\
x \equiv 6 \pmod{12} & \iff x \equiv 6 \pmod{12}.
\end{align*}
All classes modulo $12$ are represented in the right members.

If $x$ is any integer, there is some $i \in [\![0,11]\!]$ such that $x \equiv i \pmod{12}$. Therefore $x$ satisfies at least one of the congruences $x \equiv 1 \pmod 2, x \equiv 2 \pmod 3, x \equiv 0 \pmod 4, x \equiv 4 \pmod 6, x \equiv 6 \pmod{12}$.

(There are many other solutions.)
\end{proof}

Exercise 2.1.15 ($x$ satisfies at least one of the given congruences)

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Exercise 2.1.15 ($x$ satisfies at least one of the given congruences)

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