Some exercises

Here are some problems for you. Note that the problems on the Putnam exam will be more difficult than these.

1. Show that for any positive integer N (no matter however large) there exists N consecutive composite number. For exam, if N  = 3, then  14, 15, 16 is a string 3 composite numbers (not primes).

2. Show that for any non-zero real number x, the absolute value of  x + 1/x is at least 2. (show this directly without using know inequalities such as GM < = AM, etc.

3.  This is the problem of the week:   If T_n is the nth triangular number, that is T_n = n(n+1)/2, then show that the T_n can never leave a remainder of 2 or 4 when it is divided by 5.