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What is the remainder when 17^7 is divided by 35?

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17^7:35 = 11723962.0857

0.0857 x 35 = 3

The remainder is 3

  !

What is the remainder when $17^7$ is divided by $35$?

$\small{ \begin{array}{|rcll|} \hline \mathbf{17^{7} \pmod{35}} &\equiv&\left( 17\right)^{2*3+1} \pmod{35} \\ &\equiv& \left( 17^2 \right)^3*17 \pmod{35} \quad &| \quad 17^2=289 \equiv 9 \pmod{35} \\ &\equiv& \left( 9 \right)^3*17 \pmod{35} \\ &\equiv& 729*17 \pmod{35} \quad &| \quad 729 \equiv 29 \pmod{35} \\ &\equiv& 29*17 \pmod{35} \\ &\equiv& 493 \pmod{35} \quad &| \quad 493 \equiv 3 \pmod{35} \\ &\equiv& \mathbf{ 3 \pmod{35} } \\ \hline \end{array} }$

The reminder is 3