1. ## modular equations?

I am just wondering how do you solve this without a calculator 5^55mod(12)=?

2. ## Re: modular equations?

Originally Posted by ponk
I am just wondering how do you solve this without a calculator 5^55mod(12)=?

Welcome to MHF, ponk!

Here's a standard way to reduce the expression.

\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 $25 \textrm{ mod }12 = 1$, but otherwise the process could have been repeated.

For more complicated expressions Euler's theorem comes in handy.

thx a bunch