Homework Help: Proof of divisibility

1. Sep 21, 2014

IDValour

1. The problem statement, all variables and given/known data

Prove that $1900^{1990} - 1$ is divisible by $1991$

2. Relevant equations

$x^n - 1 = (x - 1)(x^{n-1} + x^{n-2} + ... + 1)$

3. The attempt at a solution

Quite naturally the first step I took was to attempt the factorisation and see what that got me:

$1900^{1990} - 1 = (1900 - 1)(1900^{1989} + 1900^{1988} + ... + 1)$

And from here I somewhat fail to see where to go forward.

$1899$ is not divisible by $1991$ so do I need to work on the second part? If so I am having trouble seeing how to resolve it. It does seems possible to factorise the problem down to $(19*100)^{1990} - 1$ but then again this seems highly irrelevant.

Any help on this issue would be greatly appreciated.

2. Sep 21, 2014

Orodruin

Staff Emeritus
Try x = -1900.

Edit: Sorry, I read divisible by 1901.

3. Sep 21, 2014

IDValour

To be honest with you, I'm beginning to think that there may have been an error and that the question was intended to read as divisible by 1901 instead of 1991. Either that or with the 1990 and 1900 swapped.

4. Sep 21, 2014

Orodruin

Staff Emeritus
This might not be an unreasonable assumption. Had 1991 been a prime number you might have been able to apply Fermat's little theorem, but alas 1991 = 11*181 ...