\section*{Problem 5 - 011035}

What is the last digit of the sum \(11^{6}+12^{6}+13^{6}+14^{6}+15^{6}+16^{6}\)?

Justify your statement!
}

It holds:

\[
\begin{array}{r}
11^{6} \equiv 1^{6} \equiv 1(\bmod 10) \\
12^{6} \equiv 2^{6} \equiv 64 \equiv 4(\bmod 10) \\
14^{6} \equiv 4^{6} \equiv 16^{3} \equiv 6^{3} \equiv 6(\bmod 10)(\text{since} 6 \cdot 6 \equiv 6(\bmod 10)) \\
13^{6} \equiv 3^{6} \equiv 27 \cdot 27 \equiv 49 \equiv 9(\bmod 10) \\
15^{6} \equiv 5^{6} \equiv 5(\bmod 10)(\text{since} 5 \cdot 5 \equiv 5(\bmod 10)) \\
16^{6} \equiv 6^{6} \equiv 6(\bmod 10)(\text{since} 6 \cdot 6 \equiv 6(\bmod 10))
\end{array}
\]

Thus, \(11^{6}+12^{6}+13^{6}+14^{6}+15^{6}+16^{6} \equiv 1+4+9+6+5+6 \equiv 1(\bmod 10)\).