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Find the remainder when $3 \times 13 \times 23 \times 33 \times \ldots \times 183 \times 193$ is divided by $5$.

 Apr 24, 2019
 #1
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product(3, 13, 23 , 33 , 43 , 53 , 63 , 73 , 83 , 93 , 103 , 113 , 123 , 133 , 143 , 153 , 163 , 173 , 183 , 193)  mod 5

 = 1  

 Apr 25, 2019
 #2
avatar+22515 
+2

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

 

laugh

 Apr 25, 2019
 #3
avatar+30 
0

Thanks!

er1004  Apr 25, 2019

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