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# What is the remainder of 5^2010 when it is divided by 7?

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What is the remainder of 5^2010 when it is divided by 7?

Aug 9, 2017

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5^2010 when it is divided by 7?

5^2010 mod 7 =1 The remainder.

Aug 9, 2017
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What is the remainder of 5^2010 when it is divided by 7?

$$\begin{array}{rcll} 5^{2010} \pmod {7} &=& \ ? \end{array}$$

$$\small{ \begin{array}{|lrcll|} \hline 1. & gcd(7,5) &=& 1 \qquad | \qquad 7 \text{ and } 5 \text{ are relatively prim } \\ 2. & 7 \text{ is a prim number } \\ 3. & \phi() \text{ is Euler's totient function, Euler's phi function }\\ & \phi(p) &=& p-1 \qquad p \text{ is a prim number} \\ & \phi(7) &=& 7-1 \\ & \phi(7) &=& {\color{red}6} \\ 4. & 5^{\phi(7)} &\equiv& 1 \pmod{7} \\ & 5^{{\color{red}6}} &\equiv& 1 \pmod{7} \\ \hline &\text{ Let } \phi(n) \text{ denote the totient function. } \\ &\text{Then } a^{\phi(n)} \equiv 1 \pmod {n} \text{ for all } a \text{ relatively prime to } n. \\ \hline \end{array} }$$

$$\begin{array}{lrcll} 5. & 2010 &=& {\color{red}6}\cdot 335 \\ & 5^{2010 } \pmod{7} &=& 5^{ {\color{red}6}\cdot 335 } \pmod{7} \\ & &=& ( 5^{ {\color{red}6} } )^{335} \pmod{7} \quad & | \quad 5^{{\color{red}6}} \equiv {\color{blue}1} \pmod{7} \\ & &\equiv& ( {\color{blue}1} )^{335} \pmod{7} \\ & &\equiv& 1 \pmod{7} \\ \end{array}$$

The remainder of $$\frac{5^{2010}} {7}$$ is $$\mathbf{1}$$

Aug 9, 2017