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Number Theory

Basic Applications of Modular Arithmetic

Basic Applications of Modular Arithmetic: Level 3 Challenges

         

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

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

Find the remainder when \(a\) is divided by 13.


Try this set RMO Practice Problems

Find the last three digits of the number

\[ 3 \times 7 \times 11 \times 15 \times \cdots \times 2003. \]

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

\(\)
Details and Assumptions:

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

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

Details and Assumptions:

  • Here GCD denotes greatest common divisor.
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