Prove the following statement using mathematical induction. If n E N, then 7" - 4" is divisible by 3. Proof by Induction: Q. E. D. A. 71 - 41 =7-4=3 =3(1), 1 E Z, thus 3 is divisible by 3 and the base case holds. B. We need to show that 7*+1 - 4*+1 is divisible by 3. C. Since q, k E Z thus 7q + 4* E Z, and therefore 7*+1 - 4*+1 is divisible by 3. D. Base Case: (n = 1) E. Inductive Step: (n = k +1) F. 7*+1 _ 4*+1 = 7(7*) _ 4*+1 = 7(4* + 3q) - 4*+1 by the inductive hypothesis. = 7(4*) + 21q - 4(4*) = 21q + 4*(7 - 4) = 3(7q + 4*) G. Assume 7* - 4* is divisible by 3 for some k > 1. Thus 7* - 4* = 3q for some q e Z. H. Therefore by the principle of mathematical induction, 7" - 4" is divisible by 3 for any n E N. I. Inductive Hypothesis: (n = k)