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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### Fast modulo operation [duplicate]

Possible Duplicate: calculating $a^b \!\mod c$ I have a number of form: $p^n + p$, where $p$ is a prime number and $n$ can be any large number, for example, say $10^{12}$. What is the generic ...
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### calculate modular algorithm with large exponents [duplicate]

I don't know how to calculate the following modulo: $$321^{654} \mod 1013$$ Are there some easy way to do this?
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### How to find remainder modulo $n$, when $n$ is a large number

I am doing RSA questions and I really could use help! Can someone show me a simple way to find $25^9 \pmod{33}$?
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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 ...
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### Multiplicative inverses for elements in field

How to compute multiplicative inverses for elements in any simple (not extended) finite field? I mean an algorithm which can be implemented in software.
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### modulo of series summation

I have trouble with computing modulo. First, I have a summation of series like this: $$1+3^2+3^4+\cdots+3^n$$ And this is the formula which can be used to compute the series: S=\frac{1-3^{n+2}}{1-3^...
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### Number in tens place

A number in tens place in result of $4^{2015} \cdot 9^{2016}$ is? Obviously without using calculator, though I doubt it could count with those high numbers. By tens place I mean, for example if you ...
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### How to simpify the way of calculating $(a^b \bmod c) \bmod d$?

Right now I know the way to calculate $a^b \bmod c$ which is in the answer of the following question. Consider the case where $c$ is much larger than $d$. So, the result of $a^b \bmod c$ will be ...
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### 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.