find period of discrete cosine 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?

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 $N=250$.
