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 \(13?\)
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 \[x^2+y^2+5z^2 ?\]