Solve each of the following:

1. \(2x \equiv 1 \pmod{17}\)
2. \(3x \equiv 1 \pmod{17}\)
3. \(3x \equiv 6 \pmod{18}\)
4. \(40x \equiv 777 \pmod{1777}\)

Solve each of the following:

1. \(2x \equiv 1 \pmod{19}\)
2. \(3x \equiv 1 \pmod{19}\)
3. \(4x \equiv 6 \pmod{18}\)
4. \(20x \equiv 984 \pmod{1984}\)

Solve \(9x \equiv 4 \pmod{1453}\).

Solve the systems

1. \(x \equiv 1 \pmod{2}\), \(x \equiv 1 \pmod{3}\).
2. \(x \equiv 3 \pmod{5}\), \(x \equiv 5 \pmod{7}\), \(x \equiv 7 \pmod{11}\).
3. \(2x \equiv 1 \pmod{5}\), \(3x \equiv 2 \pmod{7}\), \(4x \equiv 3 \pmod{11}\).

When the marchers in the annual Mathematics Department Parade lined up \(4\) abreast, there was \(1\) odd person; when they tried \(5\) in a line, there \(2\) left over; and when \(7\) abreast, there were \(3\) left over. How large is the department?

Find the smallest odd \(n\), \(n > 3\), such that \(3|n\), \(5|(n+2)\), and \(7|(n+4)\).

The Fibonacci numbers are defined by \(f_{n+1}=f_n + f_{n-1}\), \(f_1=f_2=1\). Prove that \(f_{5n}\) is divisible by \(5\) for all positive integers \(n\).