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Number Theory
Basic Applications of Modular Arithmetic
Find the remainder when \(3^{247}\) is divided by \(17\).
by
Mardokay Mosazghi
\[\Large 1+\dfrac{1}{2} + \dfrac{1}{3} + \ldots + \dfrac{1}{23} = \dfrac{a}{23!}\]
Find the remainder when \(a\) is divided by 13.
Try this set RMO Practice Problems
by
Surya Prakash
Find the last three digits of the number
\[ 3 \times 7 \times 11 \times 15 \times \cdots \times 2003. \]
by
Varun Vijay
Find the smallest positive integer \(N\) such that \( 13^N \equiv 1 \pmod{2013}\).
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Details and Assumptions:
- You may choose to refer to the modulo arithmetic notation.
- 0 is not a positive integer.
by
Calvin Lin
Find the GCD of \((19! + 19, 20! + 19).\)
Details and Assumptions:
- Here GCD denotes greatest common divisor.
by
Anuj Shikarkhane