Prove via Mathematical Induction 82n 1 is divisible by 63 n

Prove via Mathematical Induction, 8^2n - 1 is divisible by 63, n>=1

Solution

Note that when n = 1

8^(2n) - 1 => 8^(2*1) - 1 => 8^2 -1 = 63

which is divisible by 63. So the statement is true when n = 1. Assume further that for some k 1, that when n = k

8^(2n) - 1 = 8^(2k) -1            ~eq 1

is divisible by 63. Let n = k + 1. Observe that

8^(2n) - 1 = 8^(2k+2) - 1 = 8^(2k)8² - 1 = 8² 8^(2k) - 8² + 8² - 1 (Adding and subtracting 8^2)

=8²(8^(2k) - 1) + 63.

Now, using eq 1 (ie. 8^(2k) -1 is divisible by 63) and by the induction hypothesis, 63 divides (8^(2k) - 1). Hence 63 divides

8²(8^(2k) - 1) + 63 => 8^(2(k+1)) - 1.
Hence Proved by Mathematical Induction that 8^2n - 1 is divisible by 63, n>=1