Question

Problem 13 (15 points). Prove that for all natural number n, 52n-1 is divisible by 8.

Answer 1

false

Explanation:

We can prove or disprove that (52n - 1) is divisible by 8 for every natural number n using mathematical induction.

Starting with the base case:

When n = 1,

(52n - 1) = ((52 · 1) - 1)

= 52 - 1

= 51

which is not divisible by 8.

Therefore, (52n - 1) is NOT divisible by 8 for every natural number n, and the conjecture is false.

Answer 2

25^n -1 is divisible by 8

Explanation:

You want a proof that 5^(2n)-1 is divisible by 8.

Expand

We can write 5^(2n) as (5^2)^n = 25^n.

Remainder

The remainder from division by 8 can be found as ...

25^n mod 8 = (25 mod 8)^n = 1^n = 1

Less 1

Subtracting 1 from 25^n mod 8 gives 0, meaning ...

5^(2n) -1 = (25^n) -1 is divisible by 8.

__

Additional comment

Let 2n+1 represent an odd number for any integer n. Then consider any odd number to the power 2k:

(2n +1)^(2k) = ((2n +1)^2)^k = (4n² +4n +1)^k

The remainder mod 8 will be ...

((4n² +4n +1) mod 8)^k = ((4n(n+1) +1) mod 8)^k

Recognizing that either n or (n+1) will be even, and 4 times an even number will be divisible by 8, the value of this expression is ...

≡ 1^k = 1

Thus any odd number to the 2n power, less 1, will be divisible by 8. The attachment show this for a few odd numbers (including 5) for a few powers.

<95141404393>