11 questions linked to/from calculating $a^b \!\mod c$
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### Modular exponentiation by hand ($a^b\bmod c$)

How do I efficiently compute $a^b\bmod c$: When $b$ is huge, for instance $5^{844325}\bmod 21$? When $b$ is less than $c$ but it would still be a lot of work to multiply $a$ by itself $b$ times, for ...
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### Proof of this result related to Fibonacci numbers: $\begin{pmatrix}1&1\\1&0\end{pmatrix}^n=\begin{pmatrix}F_{n+1}&F_n\\F_n&F_{n-1}\end{pmatrix}$?

$$\begin{pmatrix}1&1\\1&0\end{pmatrix}^n=\begin{pmatrix}F_{n+1}&F_n\\F_n&F_{n-1}\end{pmatrix}$$ Somebody has any idea how to go about proving this result? I know a proof by ...
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### exponential equation

$$\sqrt{(5+2\sqrt6)^x}+\sqrt{(5-2\sqrt6)^x}=10$$ So I have squared both sides and got: $$(5-2\sqrt6)^x+(5+2\sqrt6)^x+2\sqrt{1^x}=100$$ $$(5-2\sqrt6)^x+(5+2\sqrt6)^x+2=100$$ I don't know what to do ...
### Calculate $7^{154} \pmod{341}$ [duplicate]
how to calculate remainder of $7^ {154}$ when it is divided by $341$. Could you please state which method or theorem to use.