+30 Find the remainder when $3 \times 13 \times 23 \times 33 \times \ldots \times 183 \times 193$ is divided by $5$.
product(3, 13, 23 , 33 , 43 , 53 , 63 , 73 , 83 , 93 , 103 , 113 , 123 , 133 , 143 , 153 , 163 , 173 , 183 , 193) mod 5
= 1
+26367 Find the remainder when \(3 \times 13 \times 23 \times 33 \times \ldots \times 183 \times 193\) is divided by \(5\).
\(\begin{array}{|rcll|} \hline 3 \pmod{5} &\equiv& 3 \\ 13 \pmod{5} &\equiv& 3 \\ 23 \pmod{5} &\equiv& 3 \\ 33 \pmod{5} &\equiv& 3 \\ \ldots \\ 183 \pmod{5} &\equiv& 3 \\ 193 \pmod{5} &\equiv& 3 \\ \hline \end{array}\)
\(\begin{array}{|rcll|} \hline && \mathbf{3 \times 13 \times 23 \times 33 \times \ldots \times 183 \times 193 \pmod{5}} \\ &\equiv& 3 \times 3 \times 3 \times 3 \times \ldots \times 3 \times 3 \pmod{5} \\ &\equiv& 3^{20} \pmod{5} \quad | \quad 3^2\pmod{5} \equiv 9 \pmod{5} \equiv -1 \pmod{5} \\ &\equiv& \left(3^2\right)^{10} \pmod{5} \\ &\equiv& \left(-1\right)^{10} \pmod{5} \\ & \mathbf{\equiv}& \mathbf{1 \pmod{5}} \\ \hline \end{array}\)
The remainder is 1
