Number Theory

Basic Applications of Modular Arithmetic

Modular Arithmetic - Problem Solving

         

What is the remainder of 17762011!\displaystyle {1776}^{2011!} upon division by 2000?2000?

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

What is the remainder of 14410\displaystyle {144}^{10} upon division by 13?13?

What is the remainder of mm satisfying 11+12+13++133=m33!\frac{1}{1}+\frac{1}{2}+\frac{1}{3}+\cdots+\frac{1}{33}=\frac{m}{33!} upon division by 17?17 ?

For three integers x,x, yy and z,z, which of the following can NOT be expressed as x2+y2+5z2?x^2+y^2+5z^2 ?

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