Linked Questions

7
votes
5answers
5k views

calculating $a^b \!\mod c$ [duplicate]

What is the fastest way (general method) to calculate the quantity $a^b \!\mod c$? For example $a=2205$, $b=23$, $c=4891$.
2
votes
4answers
18k views

Find the remainder of a number with a large exponent [duplicate]

I have to find the remainder of $10^{115}$ divided by 7. I was following the way my book did it in an example but then I got confused. So far I have, $\overline{10}^{115}$=$\overline{10}^{7*73+4}$=($...
3
votes
4answers
931 views

What is the best algorithm for finding the last digit of an enormous exponent? [duplicate]

I found most answers here not clear enough for my case such as $$ 123155131514315^{4515131323164343214547} $$ I wrote the $n\bmod10$ in Python and execution time ran out. So I need a faster ...
3
votes
5answers
444 views

What is $26^{32}\bmod 12$? [duplicate]

What is the correct answer to this expression: $26^{32} \pmod {12}$ When I tried in Wolfram Alpha the answer is $4$, this is also my answer using Fermat's little theorem, but in a calculator ...
3
votes
4answers
2k views

How to compute $3^{2003}\pmod {99}$ by hand? [duplicate]

Compute $3^{2003}\pmod {99}$ by hand? It can be computed easily by evaluating $3^{2003}$, but it sounds stupid. Is there a way to compute it by hand?
4
votes
4answers
638 views

Find $6^{1000} \mod 23$ [duplicate]

Find $6^{1000} \mod 23 $ Having just studied Fermat's theorem I've applied $6^{22}\equiv 1 \mod 23 $, but now I am quite clueless on the best way to proceed. This is what I've tried: Raising ...
1
vote
3answers
4k views

Find the remainder $4444^{4444}$ when divided by 9 [duplicate]

Find the remainder $4444^{4444}$ when divided by 9 When a number is divisible by 9 the possible remainder are $0, 1, 2,3, 4,5,6,7,8$ we know that $0$ is not a possible answer. My friend told me the ...
0
votes
2answers
1k views

How to calculate $5^{2003}$ mod $13$ [duplicate]

How to calculate $5^{2003}$ mod $13$ using fermats little theorem 5^13-1 1 mod 13 (5^12)^166+11 mod 13 a+b modn=(a modn + b modn) modn (1+11mod13)mod13 12 mod 13 = 12 why answer is 8 ? how ...
1
vote
2answers
5k views

Find $11^{644} \mod 645$ [duplicate]

Can someone just explain to me the basic process of what is going on here? I understand everything until we start adding 1's then after that it all goes to hell. I just need some guidance. The Problem ...
3
votes
4answers
520 views

Find remainder when $777^{777}$ is divided by $16$ [duplicate]

Find remainder when $777^{777}$ is divided by $16$. $777=48\times 16+9$. Then $777\equiv 9 \pmod{16}$. Also by Fermat's theorem, $777^{16-1}\equiv 1 \pmod{16}$ i.e $777^{15}\equiv 1 \pmod{16}$. ...
1
vote
3answers
932 views

How do you calculate a large power modulo a small number? [duplicate]

How do I calculate $12345^{12345} \operatorname{mod} 17$? I cant do it on a calculator? How would I show this systematically?
2
votes
1answer
2k views

Finding remainder of dividing and moding large exponents [duplicate]

Can anyone explain how to calculate the remainder for types of problems like this: $2^{2131312213123}$ divided by 100 $13^{6601}$ mod 77
0
votes
4answers
418 views

How to find remainder when $2^{2018}$ is divided by 43? [duplicate]

What is the remainder when $2^{2018}$ is divided by $43$? I know that this has something to do with one of Fermat's Theorems. I am almost at a loss as to how to solve for the remainder or why it has ...
4
votes
5answers
110 views

Find the last two digits of $1717^{1717}$ [duplicate]

$1717^{1717} \mod 100$ Since $\phi(100) = 40$ , we can transform this into: $17^{37}101^{37} \mod 100 = 17^{37} \mod 100$ How do I proceed further?
2
votes
2answers
219 views

Find the remainder when $2^{2016}$ is divided by $47$ [duplicate]

What is the remainder when $2^{2016}$ is divided by $47$? I have done Fermat's little theorem and I have now this: $2^{2016} \equiv 2^{38} \pmod{47}$ My issue is that $2^{38}$ is too large of a ...
0
votes
2answers
203 views

Discrete Mathematics help: $2017^{2017}$ mod $13$ [duplicate]

I'm not too sure how I would go about solving $2017^{2017}$ mod $13$.
1
vote
6answers
676 views

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.
0
votes
3answers
240 views

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
2
votes
2answers
822 views

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'...
0
votes
1answer
133 views

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 ...
2
votes
3answers
117 views

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 ...
2
votes
4answers
131 views

Find $ \ \ 8^{504} \equiv \pmod 5$ [duplicate]

Find $ \ \ 8^{504} \equiv \pmod 5 $ Answer: $ 8^{504} \equiv 2^{1512} \pmod 5 $ Now , $$ \begin{align}2^4 &\equiv 1 \pmod 5 \\\text{or, } \left(2^{4}\right)^{378} &\equiv 1^{378} \pmod 5 \...
0
votes
2answers
92 views

Remainder when $4^{2018}$ is divided by $29$ [duplicate]

Find the remainder after division when $4^{2018}$ is divided by $29$. My approach: we have $$4^{2018}=16^{1009}=-\left(1-17\right)^{1009}=-\left(\binom{1009}{0}-\binom{1009}{1}17+\binom{1009}{2}17^2+...
1
vote
3answers
114 views

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 ?
0
votes
3answers
353 views

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 ...
0
votes
3answers
77 views

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.
1
vote
6answers
160 views

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.....
1
vote
2answers
115 views

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. ...
2
votes
5answers
104 views

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 ...
2
votes
1answer
342 views

Remainder when dividing $13^{3530}$ with $12348$ [duplicate]

Find the remainder when dividing $13^{3530}$ with $12348$. How do I solve these type of exercises? I know there's some algorithm for solving them, I just haven't found a concrete example. Could ...

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