find period of discrete cosine

Question

let us consider following

we should find period of this discrete signal,for periodicity we should have

$x[n+kN]=x[n]$

or $10\cos(0.088\pi(n+kN) +\phi)=10\cos(0.088\pi n+\phi)$

or

$0.088\pi kN=2\pi m$

from there it is clear that $0.088*k*N=2*m$

or $ kn=22.72727272727273m$. How can I continue?

Answer 1

For $k=1$ you have

$$N=\frac{2}{0.088}m$$

Now you need to choose the smallest $m$ such that $N$ is an integer:

$$N=\frac{2000}{88}m=\frac{250}{11}m$$

The fraction cannot be reduced anymore, so you must choose $m=11$, which results in $N=250$.