I am just wondering how do you solve this without a calculator 5^55mod(12)=?
Thx in advance
Welcome to MHF, ponk!
Here's a standard way to reduce the expression.
$\displaystyle \begin{align*} 5^{55} \textrm{ mod }12 &=5^{1+2 \times 27} \textrm{ mod }12 \\ &=5 \times (5^2)^{27} \textrm{ mod }12 \\ &= 5 \times (25 \textrm{ mod }12)^{27} \textrm{ mod }12 \\ &= 5 \times (1)^{27} \textrm{ mod }12 \\ &= 5 \end{align*} $
In this case we were done rather quickly since $\displaystyle 25 \textrm{ mod }12 = 1$, but otherwise the process could have been repeated.
For more complicated expressions Euler's theorem comes in handy.