Number Theory
# Basic Applications of Modular Arithmetic

What is the remainder of $\displaystyle {1776}^{2011!}$ upon division by $2000?$

What is the remainder of ${22}!$ upon division by $23?$

What is the remainder of $\displaystyle {144}^{10}$ upon division by $13?$

For three integers $x,$ $y$ and $z,$ which of the following can **NOT** be expressed as $x^2+y^2+5z^2 ?$