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.