I. Determine the remainder when 14^(1000) is divided by 13

2. Determine the remainder when 12^(1001) is divided by 13.

3. Determine the remainder when 5'^(30) is divided by 31.

4. Show that 2(5n+1) + 5(n+2) is divisible by 27 for any positive integer n.

(use modular arithmetic if possible)