Number Theory - MOD Basic Basic

Video tutorial

Lecture Notes

Every competition participant should learn modular arithmetic, or MOD in short. This is because a majority of number theory problems are MOD related. This tutorial covers MOD basics. A separate course will cover more advanced topics.


Students should be familiar with the regular arithmetic operations (addition, subtraction, multiplication, division, and exponentiation) and


  • MOD definition
  • Basic operations and properties
  • Evaluate MOD expressions and related techniques (e.g. negative one technique etc)
  • Solve sum of digits problems (The MOD by 9 technique)
  • Solve ending digits problem



(4157) What is the tens digit of $321^{123}$?

(4158) Find the last two digits of $123^{321}$.

(4159) Determine the last two digits of $312^{123}$.

(1482) What is the hundreds digit of $2011^{2011}?$

(2216) Let $f(n)$ denote the sum of the digits of $n$. Find $f(f(f(4444^{4444})))$.

More Practice Problems click here >>>