Number Theory

Basic Applications of Modular Arithmetic

Basic Applications of Modular Arithmetic: Level 3 Challenges

Find the remainder when 3 247 3^{247} is divided by 17 17 .

Mardokay Mosazghi byMardokay Mosazghi

1 + 1 2 + 1 3 + … + 1 23 = a 23 ! \Large 1+\dfrac{1}{2} + \dfrac{1}{3} + \ldots + \dfrac{1}{23} = \dfrac{a}{23!}

Find the remainder when a a is divided by 13.

Try this setRMO Practice Problems
Surya Prakash bySurya Prakash

Find the last three digits of the number

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

Varun Vijay byVarun Vijay

Find the smallest positive integer N N such that 1 3 N ≡ 1 ( m o d 2013 ) 13^N \equiv 1 \pmod{2013} .


Details and Assumptions:

  • You may choose to refer to themodulo arithmeticnotation.
  • 0 is not a positive integer.
Calvin Lin byCalvin Lin

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


Note:GCD stands for the greatest common divisor.

Anuj Shikarkhane byAnuj Shikarkhane
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