- #1
Math100
- 780
- 220
- Homework Statement
- For ## n\geq 1 ##, use congruence theory to establish the following divisibility statement:
## 7\mid (5^{2n}+3\cdot 2^{5n-2}) ##.
- Relevant Equations
- None.
Proof:
Let ## n\geq 1 ## be a natural number.
Note that ## 5^{2}\equiv 4\pmod 7\implies (5^{2})^{n}\equiv 4^{n}\pmod {7} ##.
Now observe that \begin{align*} (3\cdot 2^{5n-2})&\equiv (3\cdot 2^{3}\cdot 2^{5n-5})\pmod {7}\\
&\equiv [3\cdot 2^{3}\cdot (2^{5})^{n-1}]\pmod {7}\\
&\equiv [24\cdot (2^{5})^{n-1}]\pmod {7}\\
&\equiv (3\cdot 4^{n-1})\pmod {7}\\
&\equiv (-4\cdot 4^{n-1})\pmod {7}\\
&\equiv -(4^{n})\pmod {7}.
\end{align*}
Thus ## 5^{2n}+3\cdot 2^{5n-2}\equiv (4^{n}-4^{n})\pmod 7\equiv 0\pmod 7 ##.
Therefore, ## 7\mid (5^{2n}+3\cdot 2^{5n-2}) ## for ## n\geq 1 ##.
Let ## n\geq 1 ## be a natural number.
Note that ## 5^{2}\equiv 4\pmod 7\implies (5^{2})^{n}\equiv 4^{n}\pmod {7} ##.
Now observe that \begin{align*} (3\cdot 2^{5n-2})&\equiv (3\cdot 2^{3}\cdot 2^{5n-5})\pmod {7}\\
&\equiv [3\cdot 2^{3}\cdot (2^{5})^{n-1}]\pmod {7}\\
&\equiv [24\cdot (2^{5})^{n-1}]\pmod {7}\\
&\equiv (3\cdot 4^{n-1})\pmod {7}\\
&\equiv (-4\cdot 4^{n-1})\pmod {7}\\
&\equiv -(4^{n})\pmod {7}.
\end{align*}
Thus ## 5^{2n}+3\cdot 2^{5n-2}\equiv (4^{n}-4^{n})\pmod 7\equiv 0\pmod 7 ##.
Therefore, ## 7\mid (5^{2n}+3\cdot 2^{5n-2}) ## for ## n\geq 1 ##.
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