A postal system uses only 5 and 7 cent stamps. Show that every quantity of postage...

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A postal system uses only 5 and 7 cent stamps. Show that every quantity of postage 24 cents and higher can be made from these stamps. Show that for any positive integer n there exists a nonnegative integer k such that n is one of 3k, 3k + 1,3k-1. What is wrong with the following 'proof that all horses are the same color? Let P(n) be the statement that any n horses have the same color. The base case P(1) holds because any horse has the same color as itself. For the inductive step, assume P(k) holds. Consider any k + 1 horses. The first k have the same color by the inductive hypothesis, as do the last k, and these two sets have k-1 horses in common so all k have the same color. Hence by induction all horses have the same color. A postal system uses only 5 and 7 cent stamps. Show that every quantity of postage 24 cents and higher can be made from these stamps. Show that for any positive integer n there exists a nonnegative integer k such that n is one of 3k, 3k + 1,3k-1. What is wrong with the following 'proof that all horses are the same color? Let P(n) be the statement that any n horses have the same color. The base case P(1) holds because any horse has the same color as itself. For the inductive step, assume P(k) holds. Consider any k + 1 horses. The first k have the same color by the inductive hypothesis, as do the last k, and these two sets have k-1 horses in common so all k have the same color. Hence by induction all horses have the same color.