modular arithmetics - remainder 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?
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!
Try codepad.org

python works with the big number correctly

http://codepad.org/mZ9HW4zW