Problem
Solve $f(x) = 6x^3 + 27x^2 + 17x + 20 \equiv 0 \pmod{30}$
My attempt was:
Since $30 = 2.3.5$, we then have:
$$ \begin{cases} f(x) \equiv 0 \pmod{2}\\ f(x) \equiv 0 \pmod{3}\\ f(x) \equiv 0 \pmod{5}\\ \end{cases} $$
By inspection, we see that: $$ \begin{cases} x \equiv 0 \pmod{2}\\ x \equiv 1 \pmod{2}\\ x \equiv 2 \pmod{3}\\ x \equiv 0 \pmod{5}\\ x \equiv 1 \pmod{5}\\ \end{cases} $$
Hence, there will be four cases:
Case 1 $$ \begin{cases} x \equiv 0 \pmod{2}\\ x \equiv 2 \pmod{3}\\ x \equiv 1 \pmod{5}\\ \end{cases} $$ Case 2
$$ \begin{cases} x \equiv 0 \pmod{2}\\ x \equiv 2 \pmod{3}\\ x \equiv 0 \pmod{5}\\ \end{cases} $$ Case 3 $$ \begin{cases} x \equiv 1 \pmod{2}\\ x \equiv 2 \pmod{3}\\ x \equiv 1 \pmod{5}\\ \end{cases} $$ Case 4 $$ \begin{cases} x \equiv 1 \pmod{2}\\ x \equiv 2 \pmod{3}\\ x \equiv 0 \pmod{5}\\ \end{cases} $$
Apply Chinese Remainder Theorem for the four system of equation, where $M = 30$:
$$ \begin{cases} M_1 = \frac{30}{2} = 15\\ M_2 = \frac{30}{3} = 10\\ M_3 = \frac{30}{5} = 6\\ \end{cases} $$ And, $$ \begin{cases} 15y_1 \equiv 1 \pmod{2} \implies y_1 = 1\\ 10y_2 \equiv 1 \pmod{3} \implies y_2 = 1\\ 6y_3 \equiv 1 \pmod{5} \implies y_3 = 1\\ \end{cases} $$
Therefore, the four solutions are: $$ \begin{cases} x_1 = 1.15.0 + 1.10.2 + 1.6.1 = 26\\ x_2 = 1.15.0 + 1.10.2 + 1.6.0 = 20\\ x_3 = 1.15.1 + 1.10.2 + 1.6.1 = 41\\ x_4 = 1.15.1 + 1.10.2 + 1.6.0 = 35\\ \end{cases} $$
Edit
Add missing cases suggested by yunone
Case 5 $$ \begin{cases} x \equiv 0 \pmod{2}\\ x \equiv 2 \pmod{3}\\ x \equiv 2 \pmod{5}\\ \end{cases} $$ Case 6 $$ \begin{cases} x \equiv 1 \pmod{2}\\ x \equiv 2 \pmod{3}\\ x \equiv 2 \pmod{5}\\ \end{cases} $$
And the last two solutions are: $$ \begin{cases} x_5 = 1.15.0 + 1.10.2 + 1.6.2 = 32 \equiv 2 \pmod{30}\\ x_6 = 1.15.1 + 1.10.2 + 1.6.2 = 47 \equiv 17 \pmod{30}\\ \end{cases} $$
Am I in the right track? Any idea?
Thanks,