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(29^13 - 5^13) mod 7 = 3

Compute $29^{13} - 5^{13}$ modulo 7

Compute $29^{13} - 5^{13}$ modulo 7

$\begin{array}{|rcll|} \hline && \mathbf{29^{13} - 5^{13} \pmod{ 7}} \quad & | \quad \mathbf{29} \equiv {\color{red}1} \pmod{7} \\ &\equiv& {\color{red}1} ^{13} - 5^{13} \pmod{ 7} \\ &\equiv& 1 - 5^{13} \pmod{ 7} \quad & | \quad \mathbf{5} \equiv {\color{red}-2} \pmod{7} \\ &\equiv& 1 - \left({\color{red}-2}\right)^{13} \pmod{ 7} \quad & | \quad 2^{13} = 8192 \\ &\equiv& 1 - (-8192) \pmod{ 7} \\ &\equiv& 1 +8192 \pmod{ 7} \\ &\equiv& 8193 \pmod{ 7} \\ &\mathbf{\equiv}& \mathbf{3 \pmod{ 7}} \\ \hline \end{array}$