Number Theory
# Basic Applications of Modular Arithmetic

Find the remainder when $70!$ is divided by $5183$.

Note: Don't use a computational device!

$\large \displaystyle\sum_{k=0}^{1007!+1}{10^k}$

Find the remainder when the summation above is divided by the summation below.

$\large \displaystyle\sum_{k=0}^{1008}{10^k}$

Find the largest $n<10,000$ such that $\displaystyle \prod_{k=0}^{n} \binom{n}{k}$ is an odd number.

$\Huge 6^{6^{6^{6^{6^6}}}}$

Find the $6^\text{th}$ last digit from the right of the decimal representation of the above number.