180 questions linked to/from How do I compute $a^b\,\bmod c$ by hand?
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### Discrete Mathematics help: $2017^{2017}$ mod $13$ [duplicate]

I'm not too sure how I would go about solving $2017^{2017}$ mod $13$.
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### Remainder when $5^{5555}$ is divided by $10000$. [duplicate]

Find the remainder when $5^{5555}$ is divided by $10000$. A step by step guide with explanation for a beginner student in modular arithmetic is needed.
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### Remainder when $333^{333}$ is divided by $7$ [duplicate]

Find the Remainder when $333^{333}$ is divided by $7$ I think I have to find $333^{333}\equiv r \pmod7$ where $r(\ge0)$ is the remainder but how do I get in that form
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### Modular Arithmetic with Multiple Exponents [duplicate]

I understand how to do modular arithmetic on numbers with large exponents (like $8^{202}$). However, I am having trouble understanding how to calculate something like: $3^{3^{3^{3^3}}}$ mod 5 (that'...
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### Find $7^{1\,000\,000\,000\,000\,000} \bmod{107}$ [duplicate]

What is a shortcut to doing this kind of problem? I know that 7 and 107 are both prime number; thus, I assume that has something to do with the appropriate approach/solution. But beyond that I am ...
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### Calculation of $2^{XXXX} + 3^{XXXX}\pmod{11}$ [duplicate]

I've a question: How do I calculate $2^{2020} + 3^{2020}\pmod{11}$? Is there a theorem or any trick to do it? I need to show all the steps I used to calculate the Rest but I've no clue how to even ...
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### Detemine the unit digit of a number [duplicate]

Find the unit digit of the number: $$3^{7005} \times 6^{8000}$$ My turn: $$3^{7005}\times 6^{8000}=3^{7005}\times 3^{8000} \times2^{8000}$$ $$3^{13005} \times 2^{8000}$$ But i could not go on ?
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### How to compute $21^{4600} \mod 47$ [duplicate]

I am struggling to get this problem started. I have looked at similar problems in the book I am using for class (Discrete Mathematics with Applications, 7E) but none of them are seeming to help. Any ...
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### Compute $(5^{200}+7^{600}) \mod 12$ [duplicate]

Compute $(5^{200}+7^{600}) \mod 12$. I thought about somehow using the binomial theorem, but I couldn't make any progress.
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### How do I find the last two digits $2012^{2013}$? [duplicate]

How do I find the last two digits 20122013 My teacher said this was simple arithmetics(I still don't see how this is simple). I thought of using Congruence equation as 2012 is congruent to 2 mod 10.....
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### How to calculate mod of number with big exponent [duplicate]

I want to find $$5^{133} \mod 8.$$ I have noticed that $5^n \mod 8 = 5$ when $n$ is uneven and 1 otherwise, which would lead me to say that $5^{133} \mod 8 = 5$ But I don't know how to prove this. ...
### How to find remainder when $975^{40153}$ is divided by $14$? [duplicate]
I still find tricky this kind of problems. I tried to do solve it by factoring $14$ in $2*7$. Then, with Fermat's Little Theorem, I find that: $975^6\equiv 1\pmod 7$ $975^1\equiv 1\pmod 2$ How can ...
### How to find reminder of $m^{x}$ divided by $n$ using Euler's and Fermat's little theorem [duplicate]
How do you find reminder of $m^{x}$ divided by $n$ using Euler's and Fermat's little theorem? Can anyone show me step-by-step how to apply Fermat's little theorem and Euler's theorem? Example: What ...