SOLUTION: use mathematical induction to prove 7^n-3^n is divisible by 4.

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Question 773315: use mathematical induction to prove 7^n-3^n is divisible by 4.
Answer by Edwin McCravy(20056) (Show Source):
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use mathematical induction to prove 7^n - 3^n is divisible by 4.

It is true for n = 1 since 7¹ - 3¹ = 4 which is divisible by 4. We also need to show that it is true for n = 2 (which makes this proof a little different from most induction proofs as we will see.) It is true for n = 2 since 7² - 3² = 49 - 9 = 40 which is divisible by 4. Assume for some k ≧ 2 , that both 7^k - 3^k and 7^k-1 - 3^k-1 are divisible by 4. Therefore there exist integers p,q such that (A) 7^k - 3^k = 4p and (B) 7^k-1 - 3^k-1 = 4q Multiply the left side of (A) by (7 + 3) and the right side by 10 (7 + 3)(7^k - 3^k) = 40p FOIL the left side 7^k+1 - 7�3^k + 3�7^k - 3^k+1 = 40p Factor 7�3 out of the two middle terms on the left: 7^k+1 + 7�3(-3^k-1 + 7^k-1) - 3^k+1 = 40p Reverse the terms inside the parentheses: 7^k+1 + 7�3(7^k-1 - 3^k-1) - 3^k+1 = 40p By (B) above we can replace the parentheses by 4q 7^k+1 + 7�3(4q) - 3^k+1 = 40p 7^k+1 + 84q - 3^k+1 = 40p Rearrange the equation: 7^k+1 - 3^k+1 = 40p - 84q 7^k+1 - 3^k+1 = 4(10p - 21)q The right side is divisible by 4, and therefore so is the left side. QED Edwin