- #1
Math100
- 802
- 222
- Homework Statement
- For any prime ## p>3 ##, prove that ## 13 ## divides ## 10^{2p}-10^{p}+1 ##.
- Relevant Equations
- None.
Proof:
Let ## p>3 ## be any prime number.
Then ## p>3 ## is either of the form ## 6k+1 ## or ## 6k+5 ## for some ## k\in\mathbb{Z} ##.
Now we consider two cases.
Case #1: Suppose ## p=6k+1 ## for some ## k\in\mathbb{Z} ##.
Then ## 10^{2p}-10^{p}+1=10^{12k+2}-10^{6k+1}+1 ##.
Observe that
\begin{align*}
&10^{2p}-10^{p}+1\equiv [(-3)^{12k+2}-(-3)^{6k+1}+1]\pmod {13}\\
&\equiv [9((-3)^{3})^{4k}\cdot 3((-3)^{3})^{2k}+1]\pmod {13}\\
&\equiv [9(-1)^{4k}\cdot 3(-1)^{2k}+1]\pmod {13}\\
&\equiv (9+3+1)\pmod {13}\\
&\equiv 0\pmod {13}.\\
\end{align*}
Thus, ## 13\mid (10^{2p}-10^{p}+1) ##.
Case #2: Suppose ## p=6k+5 ## for some ## k\in\mathbb{Z} ##.
Then ## 10^{2p}-10^{p}+1=10^{12k+10}-10^{6k+5}+1 ##.
Observe that
\begin{align*}
&10^{2p}-10^{p}+1\equiv [(-3)^{12k+10}-(-3)^{6k+5}+1]\pmod {13}\\
&\equiv [-3((-3)^{3})^{4k+3}\cdot (-9)((-3)^{3})^{2k+1}+1]\pmod {13}\\
&\equiv [-3(-1)^{4k+3}\cdot (-9)(-1)^{2k+1}+1]\pmod {13}\\
&\equiv (3+9+1)\pmod {13}\\
&\equiv 0\pmod {13}.\\
\end{align*}
Thus, ## 13\mid (10^{2p}-10^{p}+1) ##.
Therefore, ## 13 ## divides ## 10^{2p}-10^{p}+1 ## for any prime ## p>3 ##.
Let ## p>3 ## be any prime number.
Then ## p>3 ## is either of the form ## 6k+1 ## or ## 6k+5 ## for some ## k\in\mathbb{Z} ##.
Now we consider two cases.
Case #1: Suppose ## p=6k+1 ## for some ## k\in\mathbb{Z} ##.
Then ## 10^{2p}-10^{p}+1=10^{12k+2}-10^{6k+1}+1 ##.
Observe that
\begin{align*}
&10^{2p}-10^{p}+1\equiv [(-3)^{12k+2}-(-3)^{6k+1}+1]\pmod {13}\\
&\equiv [9((-3)^{3})^{4k}\cdot 3((-3)^{3})^{2k}+1]\pmod {13}\\
&\equiv [9(-1)^{4k}\cdot 3(-1)^{2k}+1]\pmod {13}\\
&\equiv (9+3+1)\pmod {13}\\
&\equiv 0\pmod {13}.\\
\end{align*}
Thus, ## 13\mid (10^{2p}-10^{p}+1) ##.
Case #2: Suppose ## p=6k+5 ## for some ## k\in\mathbb{Z} ##.
Then ## 10^{2p}-10^{p}+1=10^{12k+10}-10^{6k+5}+1 ##.
Observe that
\begin{align*}
&10^{2p}-10^{p}+1\equiv [(-3)^{12k+10}-(-3)^{6k+5}+1]\pmod {13}\\
&\equiv [-3((-3)^{3})^{4k+3}\cdot (-9)((-3)^{3})^{2k+1}+1]\pmod {13}\\
&\equiv [-3(-1)^{4k+3}\cdot (-9)(-1)^{2k+1}+1]\pmod {13}\\
&\equiv (3+9+1)\pmod {13}\\
&\equiv 0\pmod {13}.\\
\end{align*}
Thus, ## 13\mid (10^{2p}-10^{p}+1) ##.
Therefore, ## 13 ## divides ## 10^{2p}-10^{p}+1 ## for any prime ## p>3 ##.