Number Theory

# Modular Arithmetic - Problem Solving

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 $$13?$$

What is the remainder of $$m$$ satisfying $\frac{1}{1}+\frac{1}{2}+\frac{1}{3}+\cdots+\frac{1}{33}=\frac{m}{33!}$ upon division by $$17 ?$$

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

×