Exercise 2.3.17 (Solve $x^3 - 9x^2 + 23x - 15 \equiv 0 \pmod{143}$)

Question

Solve the congruence $x^3 - 9x^2 + 23x - 15 \equiv 0 \pmod{143}$.

Richard Ganaye · Accepted Answer

\begin{proof} Watch out! $143$ is not a prime: $143 = 11 	imes 13$.

Reducing modulo $11$, we obtain, using the factorization given in Problem 16,
\begin{align*}
x^3 - 9x^2 + 23x - 15 \equiv 0 \pmod{11} 
& \iff (x-1)(x-3)(x-5) \equiv 0 \pmod {11}\
&\iff x \equiv 1,3,5 \pmod {11}.
\end{align*}
Modulo 13,
\begin{align*}
x^3 - 9x^2 + 23x - 15 \equiv 0 \pmod{13} 
& \iff (x-1)(x-3)(x-5) \equiv 0 \pmod {13}\
&\iff x \equiv 1,3,5 \pmod {13}.
\end{align*}
Therefore
\begin{align*}
&\phantom{\iff}\ x^3 - 9x^2 + 23x - 15 \equiv 0 \pmod{143} \
 & \iff x^3 - 9x^2 + 23x - 15 \equiv 0 \pmod{11} 	ext{ and } x^3 - 9x^2 + 23x - 15 \equiv 0 \pmod{13}\
&\iff x \equiv 1,3,5 \pmod {11} 	ext{ and }x \equiv 1,3,5 \pmod {13}\
&\iff 
\phantom{	ext{ or }}
\left\{
\begin{array}{ll}
x \equiv 1 \pmod {11}\
x \equiv 1 \pmod {13}
\end{array}
ight.
	ext{ or }
\left\{
\begin{array}{ll}
x \equiv 1 \pmod {11}\
x \equiv 3 \pmod {13}
\end{array}
ight.
	ext{ or }
\left\{
\begin{array}{ll}
x \equiv 1 \pmod {11}\
x \equiv 5 \pmod {13}
\end{array}
ight.\
&\phantom{\iff}\ 
	ext{ or }
\left\{
\begin{array}{ll}
x \equiv 3 \pmod {11}\
x \equiv 1 \pmod {13}
\end{array}
ight.
	ext{ or }
\left\{
\begin{array}{ll}
x \equiv 3 \pmod {11}\
x \equiv 3 \pmod {13}
\end{array}
ight.
	ext{ or }
\left\{
\begin{array}{ll}
x \equiv 3 \pmod {11}\
x \equiv 5 \pmod {13}
\end{array}
ight.\
&\phantom{\iff}\ 
	ext{ or }
\left\{
\begin{array}{ll}
x \equiv 5 \pmod {11}\
x \equiv 1 \pmod {13}
\end{array}
ight.
	ext{ or }
\left\{
\begin{array}{ll}
x \equiv 5 \pmod {11}\
x \equiv 3 \pmod {13}
\end{array}
ight.
	ext{ or }
\left\{
\begin{array}{ll}
x \equiv 5 \pmod {11}\
x \equiv 5 \pmod {13}
\end{array}
ight.\
&\iff x \equiv 1, 133, 122, 14, 3, 135, 27, 16,5 \pmod{143}.
\end{align*}
So the solutions of the congruence $ x^3 - 9x^2 + 23x - 15 \equiv 0 \pmod{143}$ are
$$ x \equiv 1,3,5,14,16,27,122,133,135 \pmod{143}.$$

\end{proof}

Exercise 2.3.17 (Solve $x^3 - 9x^2 + 23x - 15 \equiv 0 \pmod{143}$)

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Exercise 2.3.17 (Solve $x^3 - 9x^2 + 23x - 15 \equiv 0 \pmod{143}$)

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