**Edit 01**:In order to look divisibility among power divisor function where i would like to know if there a such integer $n>1 $ with y coprime to $x$ then we have: :$\sigma\_y(n)\bmod \sigma\_x(n)=0$, by [wolfram alpha](http://www.wolframalpha.com/input/?i=Table[sigma_3%28n%29+mod+sigma_2%28n%29]%3D0+%2Cn%3D2+to+5000) i have got no integer n>1 satisfies the equation $\sigma\_y(n)\bmod \sigma\_x(n)=0$ at a least it is true from $ n=2$ to $ 5000$ with $y=3$ and $x=2$,According to "Gerhard paseman Idea " we have the following question show the counter example and the problem I challenge is to find integers:$x,p, q ,y$ which presnt the counter example of the precedent equation,


Then i have tried to find these integers $x,p, q ,y$ which satisfy the following conditions but I don't succeed:


$p, q$ are primes, $x$ is a positive integer such that: 


$(01)\quad $ $b$ is not a multiple of $x$, but $(1+p^x) $ divides $(1+q^b)$. 


$(02)\quad$ $c$ is not a multiple of $x$, but $(1+q^x)$ divides $(1+p^c)$ 


$(03)\quad$ $\gcd\, (y,x)=1$, where $y={\rm lcm}\,(c,b)$ . 


**Question.** Is there an example of such integers $x,p,q,y$?


**Note**:I edited the question to show why I posted this question and to make it clear 


Thank you for any help .



Is there an example of integers ($x,p, q ,y$ ) which satisfies the below conditions in this claim?
This is possible, as expected (usually if there are no obvious reasons why not, the answer in such a problem is yes.)


Try $p=3$, $x=2$. Then we need $10|q^b+1$, $q^2+1|3^c+1$ for some odd $b,c$. First relation is possible for $q=10k-1$, $b=1$. The second holds if the order of $-3$ modulo $q^2+1$ is odd. Computations (with help of Wolframalpha) [suggest](http://www2.wolframalpha.com/input/?i=3%5E%28435%29%2B1%20modulo%20%2859%29%5E2%2B1) that $q=59$, $c=435$ work.


