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Solve integer equations, determine remainders of powers, and much more with the power of Modular Arithmetic.

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

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

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

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Find the last three digits of the number
\[ 3 \times 7 \times 11 \times 15 \times \ldots \times 2003 \]

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

**Details and assumptions**

You may choose to refer to Modulo Arithmetic notation.

0 is not a positive integer.

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

**Details and Assumptions:**

- Here
**GCD**denotes greatest common divisor.

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