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## Maths challenge 15

Posted by Chris on September 25, 2011 – 7:55 am

Prove that for integer n > 2, that 3^n-1 doesn’t divide 5^n-1 exactly.

You’ll find http://trickofmind.com/?p=957#comments useful.

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### 2 Responds so far- Add one»

1. 1. leadfarmer5 Said：

not sure how to officially prove it with a geometric proof or something, but 3 multiplied by itself any number of times will never produce a number with a 0 or 5 in the ones digit, and 5 multiplied by itself any number of times will always produce a number with a 0 or 5 in the ones digit

2. 2. Chris Said：

Hi leadfarmer5. Right now I wish that it was as simple as that. I may have bitten off more than I can chew.

I’ve number crunched up to n = 910 000 (I’ve now stopped checking), so far, only n = 1 or 2 gives an exact division. i.e.
(51-1)/(31-1) = 4/2 = 2
(52-1)/(32-1) = 24/8 = 3

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