Periodicity of sum of discrete signals In my lecture slides for school and from this website here

"The sum $z[n] = x[n] + y[n]$ of periodic signals $x[n]$ with fundamental period $N1$, and $y[n]$ with fundamental period $N2$ is periodic if the ratio of periods of the summands is rational, i.e.,
$\frac{N2}{N1}=\frac{p}{q}$
and $p$ and $q$ are integers not divisible by each other. If so, the fundamental period of $z[n]$ is $qN2 = pN1$."

I'm confused with a number of things. First if $y[n]$ and $x[n]$ were periodic, then wouldn't the ratio of their periods $\mathbf{always}$ be rational since each period is always integer? Also if each signal is not periodic then this would imply their sum could be $cos[n]+cos[n]=2cos[n]$ is not rational but the ratio of their "periods" is $1$ which is rational. So, I'm assuming that this requirement is just redundant if the original signals are periodic?
First if y[n] and x[n] were periodic, then wouldn't the ratio of their periods always be rational since each period is always integer? 


No. I mean. Yes.

$x(t)$ might be e.g. $2\pi T$-periodic, and $y(t)$ could be $5T$-periodic ($T$ being the sample period). The choice of using capital $N$ for the period was a confusing one, but signals don't need to have integer sample-multiples as periods.

BUT, and that's where I'm with you, if you say $x[n]$ is periodic, it means the sampled signal $x[n]$ is periodic – which it can't be, if the original sample would have a rational-amount-of-samples period. 

And thus, you're right: if the sampled signals are periodic, they need to have a rational ratio in periods, and thus, their sum will always also be periodic.