Number Theory

Basic Applications of Modular Arithmetic

Basic Applications of Modular Arithmetic: Level 3 Challenges


Find the remainder when 32473^{247} is divided by 1717.

1+12+13++123=a23!\Large 1+\dfrac{1}{2} + \dfrac{1}{3} + \ldots + \dfrac{1}{23} = \dfrac{a}{23!}

Find the remainder when aa is divided by 13.

Try this set RMO Practice Problems

Find the last three digits of the number

3×7×11×15××2003. 3 \times 7 \times 11 \times 15 \times \cdots \times 2003.

Find the smallest positive integer NN such that 13N1(mod2013) 13^N \equiv 1 \pmod{2013}.

Details and Assumptions:

  • You may choose to refer to the modulo arithmetic notation.
  • 0 is not a positive integer.

Find the GCD of (19!+19,20!+19).(19! + 19, 20! + 19).

Note: GCD stands for the greatest common divisor.


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