Find the least residues:

1. \(1789 \pmod{4}\)
2. \(1789 \pmod{10}\)
3. \(1789 \pmod{101}\)

If \(k \equiv 1 \pmod{4}\), then find \(6k+5 \pmod{4}\).

Show that no square has as its last digit \(2\), \(3\), \(7\), or \(8\).

What can the last digit of a fourth power be?

Show that the difference of two consecutive cubes is never divisibly by \(5\).

Show that, for \(k > 0\) and \(m \ge 1\), if \(x \equiv 1 \pmod{m^k}\), then \(x^m \equiv 1 \pmod{m^{k+1}}\).

Show that \(a^5\equiv a \pmod{10}\) for all integers \(a\).

Prove that \(6^n \equiv 1 + 5n \pmod{25}\). (Hint: induction might be useful.)