Find the orders of \(1, 2, \ldots, 12 \pmod{13}\).

One of the primitive roots of \(23\) is \(5\). Find all the others.

What are the orders of \(3, 7, 9, 11, 13, 17, 19 \pmod{20}\)? Does \(20\) have primitive roots?

Which integers have order \(6 \pmod{37}\)?

Find solutions for the following congruences, or show they don’t exist.

• \(x^2=8 \pmod{53}\)
• \(x^2=15 \pmod{31}\)
• \(x^2=54 \pmod{7}\)
• \(x^2=625 \pmod{9973}\)

Calculate \(\left(\frac{33}{73}\right)\), \(\left(\frac{34}{73}\right)\), \(\left(\frac{35}{73}\right)\), and \(\left(\frac{36}{73}\right)\).

Solve \(2x^2+3x+1 \equiv 0 \pmod{7}\) and \(2x^2+3x+1 \equiv 0 \pmod{101}\).