What is the least residue of

1. \(5^{10} \pmod{11}\)
2. \(5^{12} \pmod{11}\)
3. \(1945^{12} \pmod{11}\)

What are the last two digits of \(7^{355}\)?

What is the remainder when \(314^{162}\) is divided by \(163\)?

What is the remainder when \(314^{164}\) is divided by \(165\)?

Suppose that \(p\) is an odd prime. Show that \[1^{p-1} + 2^{p-1} + \cdots + (p-1)^{p-1} \equiv -1 \pmod{p}\] and \[1^p + 2^p + \cdots + (p-1)^p \equiv 0 \pmod{p}.\]

Show that for any two distinct primes \(p,q\),

1. \(pq\) divides \(a^{p+q} - a^{p+1} - a^{q+1} + a^2\) for all \(a\).
2. \(pq\) divides \(a^{pq} - a^{p} - a^{q} + a\) for all \(a\).

Show that if \(p\) is an odd prime, then \(2p\) divides \(2^{2p-1}-2\).