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
909 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
443 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
612 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
510 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
930 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?
1
vote
4answers
405 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 ...
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
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 ...

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