- #1
Math100
- 797
- 221
- Homework Statement
- Find the remainder when ## 4444^{4444} ## is divided by ## 9 ##.
[Hint: Observe that ## 2^{3}\equiv -1\pmod {9} ##.]
- Relevant Equations
- None.
Observe that ## 4444\equiv 7\pmod {9} ##.
This means ## 4444^{4444}\equiv 7^{4444}\pmod {9}\equiv 7^{4+40+400+4000}\pmod {9} ##.
Now we have
\begin{align*}
&7^{4}\equiv 7\pmod {9}\\
&7^{40}\equiv (7^{4})^{10}\pmod {9}\equiv 7^{10}\pmod {9}\equiv [(7^{4})^{2}\cdot 7^{2}]\pmod {9}\equiv 7^{4}\pmod {9}\equiv 7\pmod {9}\\
&7^{400}\equiv (7^{4})^{100}\pmod {9}\equiv 7^{100}\pmod {9}\equiv (7^{4})^{25}\pmod {9}\equiv 7^{25}\pmod {9}\equiv [(7^{4})^{6}\cdot 7]\pmod {9}\equiv (7^{6}\cdot 7)\pmod {9}\equiv 7\pmod {9}\\
&7^{4000}\equiv (7^{400})^{10}\pmod {9}\equiv 7^{10}\pmod {9}\equiv 7\pmod {9}.\\
\end{align*}
Thus ## 7^{4444}\equiv (7^{4000}\cdot 7^{400}\cdot 7^{40}\cdot 7^{4})\pmod {9}\equiv 7\pmod {9} ##.
Therefore, the remainder when ## 4444^{4444} ## is divided by ## 9 ## is ## 7 ##.
This means ## 4444^{4444}\equiv 7^{4444}\pmod {9}\equiv 7^{4+40+400+4000}\pmod {9} ##.
Now we have
\begin{align*}
&7^{4}\equiv 7\pmod {9}\\
&7^{40}\equiv (7^{4})^{10}\pmod {9}\equiv 7^{10}\pmod {9}\equiv [(7^{4})^{2}\cdot 7^{2}]\pmod {9}\equiv 7^{4}\pmod {9}\equiv 7\pmod {9}\\
&7^{400}\equiv (7^{4})^{100}\pmod {9}\equiv 7^{100}\pmod {9}\equiv (7^{4})^{25}\pmod {9}\equiv 7^{25}\pmod {9}\equiv [(7^{4})^{6}\cdot 7]\pmod {9}\equiv (7^{6}\cdot 7)\pmod {9}\equiv 7\pmod {9}\\
&7^{4000}\equiv (7^{400})^{10}\pmod {9}\equiv 7^{10}\pmod {9}\equiv 7\pmod {9}.\\
\end{align*}
Thus ## 7^{4444}\equiv (7^{4000}\cdot 7^{400}\cdot 7^{40}\cdot 7^{4})\pmod {9}\equiv 7\pmod {9} ##.
Therefore, the remainder when ## 4444^{4444} ## is divided by ## 9 ## is ## 7 ##.