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I need to find the remainder of ${1011}^{10}+{10}^{11}$ when divided by $101$.

According to this website it is 55 but I fail to see how.

For 1011, we can write it as $1$ because $1011=10*101+1$ so ${1011}^{10}$ is $1$.

For ${10}^{11}$, I wrote the $11$ like $8+2+1$ or $2^3+2^1+2^0$

Then multiplying the results $10\cdot100\cdot1$ and taking the remainder of 101 I got 91. $$1000=9\cdot101+91$$ So overall I have 92 for the answer, but the website says 55. Was my method or understaning wrong or is there is a problem or limitation on large numbers in that calculator?

Also is there any website or mehtod to check myself with this kind of big numbers?

Lumon
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    You did fine. I would do the $10^{11}$ part as follows: $$10^2=100\equiv-1\pmod{101},$$ so $$10^{11}=(10^2)^5\cdot10\equiv(-1)^5\cdot10=-10\equiv91\pmod{101}.$$ Looks like that website doesn't deliver. – Jyrki Lahtonen Nov 09 '16 at 20:38
  • Even simpler than what I did, thanks a lot for showing me this way and for the answer. – Lumon Nov 09 '16 at 20:41

3 Answers3

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I don't think it's the web-site's fault. The input on that page was 1.1156078e30. You're not giving it the correct number $1011^{10} + 10^{11}$, which is 1115607835569227940375059334601, just the first $8$ decimal digits. All the digits matter here!

Robert Israel
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$$1011^{10}+10^{11}\equiv (101\times 10+1)^{10}+10\times 100^5\equiv(0\times10 +1)^{10}+10\times(-1)^5\equiv1-10\equiv92\mod101$$

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Try codepad.org

python works with the big number correctly

http://codepad.org/mZ9HW4zW

kotomord
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