I was doing a proof by induction and stepped over something, which I intuitively thought it has to be wrong. But I better ask:
proof by induction that this is divisible by 15:
$$ 2^{4n}-1 $$
i)Basis n=1
ii) assume n=k works
iii) try out n=k+1
$$ i) 2^4-1:15=1 \\ii) 2^{4k}-1=15q \\iii) 2^{4(k+1)}-1=2^{4}*2^{4k}-1$$
And this is where I was confused and I thought that this must be illegal: May I substitute $$15q=2^{4k}-1$$ here?
In my opinion this is forbidden since the order of operation would be violated.
I rather did the following:
$$15q=2^{4k}-1 \space |+1 \\15q+1=2^{4k} \\=> iii) 2^4(15q+1)-1\\=16*15q-15\\=(16q-1)*15=>divisible \space by \space 15$$