PROBLEM SET 1

1. What are the possible digits that the number 4^n can have as the last digit? 
Write down the exact set of numbers that can be attained (recall that in class we saw that for 3^n, the last digit 
was one of 1,3,7 and 9. Prove your claim using the principle of simple induction. Give special attention to the 
structure of your answer.

2. Show inductively the the numbers of pairs of different numbers in {1,2..,n} is n*(n-1)/2 for every natural n.
Notice that the pairs are unordered, which means that the {1,3} is the same as {3,1}. 

 Example, There are 6 = 4*(4-1)/2 pairs in {1,2,3,4} :
  {1,2}, {1,3},{1,4},{2,3},{2,4},{3,4}