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

Question 773315: use mathematical induction to prove 7^n-3^n is divisible by 4.
Answer by Edwin McCravy(20056) (Show Source): You can put this solution on YOUR website!
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

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