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# What is the remainder when 333^{333} is divided by 11?

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What is the remainder when 333^{333} is divided by 11?

Aug 14, 2018

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What is the remainder when 333^{333} is divided by 11?

$$\begin{array}{|rcll|} \hline && 333^{333} \pmod{11} \quad & | \quad 333 \equiv 3 \pmod{11} \\ &\equiv & 3^{333} \pmod{11} \quad & | \quad \text{Fermat's little theorem } 3^{10} \equiv 1 \pmod{11} \\ &\equiv & 3^{10\cdot 33+3} \pmod{11} \\ &\equiv & \left(3^{10}\right)^{33} 3^3 \pmod{11} \\ &\equiv & (1)^{33} 3^3 \pmod{11} \\ &\equiv & 1\cdot 3^3 \pmod{11} \\ &\equiv & 3^3 \pmod{11} \\ &\equiv & 27 \pmod{11} \\ &\mathbf{\equiv} & \mathbf{5 \pmod{11}} \\ \hline \end{array}$$

Aug 14, 2018