Number Theory
# Basic Applications of Modular Arithmetic

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.

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 \( 13^N \equiv 1 \pmod{2013}\).

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**Details and Assumptions:**

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

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

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**Note:** GCD stands for the greatest common divisor.

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