The followings from the text 7.3:29, 31 7.4:18, 19 Example 1: Let S be a set of 6 positive integers whose maximum is at most 14. Show that the sums of the elements in all the non-empty subsets of S cannot all be distinct: Sol: There are 2^6-1=63 non-empty subsets and the sum of every subset of S is at most 9+10+...+14=69. So the pigeonhole pricincple cannot be applied directly. Instead, let's consider subsets of size at most 5. Their total sum is at most 10+11+..+14=60. There are 62 non-empty subsets with at most 5 elements. So at least two of them have equal sum. Example 2: Let m be an odd positive integer. Prove that there is a positive inteer n such that m divides 2^n-1. Sol: Consider the m+1 positive integers 2^1-1, 2^2-1, ..., 2^m-1, 2^{m+1}-1. By PHP, there are positive integers 1<= s