I was kind of lost with the following example of induction: $ (11^{n+1} + 12^{2n-1}) \mathbin{\%} 133 = 0 $
It shows the following steps to solve it: (I excluded base proof for n = 1)
$ 11^{(n+1)+1} + 12^{2\cdot(n+1)-1} = $ $ = 11 \cdot 11^{n+1} + 12^{2} \cdot 12^{2n-1} = $ $ = 11 \cdot 11^{n+1} + 11 \cdot 12^{2n-1} + 133 \cdot 12^{2n-1} = $ $ = 11 \cdot ( 11^{n+1} + 11 \cdot 12^{2n-1}) + 133 \cdot 12^{2n-1} $ $ = 11 \cdot 133k + 133 \cdot 12^{2n-1} = $ $ = 133 (11k + 12^{2n-1}) $
I prooven for n=1, but i am totally confused by what follows next in this example there is no description in the book why are these steps taken whats going on etc.
I get the first line expanding to n+1 but i dont know how he gets $ 12^{2} $ from factoring that. And i dont get the rest of it.
Thanks in advance for any help.