NL-NWO 2010 Problem 2

A number is called polite if it can be written as $m + (m+1)+...+ n$ , for certain positive integers $m <n$ . For example: $18$ is polite, since $18 =5 + 6 + 7$ . A number is called a power of two if it can be written as $2^{\ell}$ for some integer $\ell \ge 0$ . (a) Show that no number is both polite and a power of two. (b) Show that every positive integer is polite or a power of two.