1. Show that for any positive integer n, there exists a positive integer m such that n divides m (namely, m = kn for some integer k), and the decimal digits of m are only O and 1. For example, n = 7, m? = 1001. In addition to show that this is true in general, use n = 7 and n = 12 to show how to find such m.

6. The girth of a graph G is the length of the shortest cycle in G. Let G be a simple graph with y vertices, e edges, and girth g. It is known that if G is planar, then e . Let K5 be a complete graph with 5 vertices, K5and K3,3 be a complete bipartite graph with 3 vertices in each partition. Draw Ks and K3,3, and show that they are not planar.

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