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 May 16th, 2017, 05:41 PM #1 Newbie   Joined: May 2017 From: Trinidad Posts: 1 Thanks: 0 Induction Prove that, for all positive integers n,3^2n - 1 is divisible by 8
May 16th, 2017, 06:08 PM   #2
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Quote:
 Originally Posted by Qwane Prove that, for all positive integers n,3^2n - 1 is divisible by 8
Easier by modular arithmetic. $3^{2n} = (3^2)^n \equiv 1^n \equiv 1 \pmod 8$. Did you make any progress on your induction?

Last edited by Maschke; May 16th, 2017 at 06:13 PM.

May 17th, 2017, 02:10 AM   #3
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Quote:
 Originally Posted by Qwane Prove that, for all positive integers n,3^2n - 1 is divisible by 8
I take it you mean that, for any positive integer n, 3^(2n) - 1 is divisible by eight, not that 3^2n - 1 is divisible by eight. The latter statement is generally false.

$3^2 * 2 - 1 = 9 * 2 - 1 = 17.$ Not divisible by 8.

$3^2 * 3 - 1 = 9 * 3 - 1 = 26.$ Not divisible by 8.

There are two overall steps in a proof by induction.

What is the first one?

What did you get in that step?

What is the second step?

The second step is always where the difficulty lies. How do you initiate that step?

Last edited by JeffM1; May 17th, 2017 at 02:14 AM.

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