If $792$ divides the integer $13xy45z$, find the digits $x,y$ and $z$.
I know that i have to use some divisibility test but i am stuck how to use it and solve the above example.
Note that $792=8\times 9\times 11$.
A number is divisible by $8$ if and only if the last three digits are divisible by $8$, so we need $45z$ to be divisible by $8$. That will give you the value of $z$.
A number is divisible by $9$ if and only if the sum of the digits is divisible by $9$. So you need $1+3+x+y+4+5+z$ to be a multiple of $9$. This means that $4+x+y+z$ must be a multiple of $9$. You will already know the value of $z$, so this gives you information about $x+y$.
A number is a multiple of $11$ if and only if the sum of the odd-place digits minus the sum of the even-place digits is a multiple of $11$. So we need $1-3+x-y+4-5+z$ to be a multiple of $11$. You will already have information about $z$, so this will give you information about $x-y$.
From knowing stuff about $x+y$ and about $x-y$, you should be able to figure out both $x$ and $y$.