Q:

The five digit number $xy26z$ (each of the letters represents a digit) is divisible by $8$, $9$ and $11$. What is the value of $x$? 

The only thing I came up with was using the divisibility criteria, so:
$$8 + z \equiv 0 \pmod{8} \implies z \equiv 0 \pmod 8$$
$$x + y + 8 + z \equiv 0 \pmod{9}$$
$$x - y - 4 + z \equiv 0 \pmod{11}$$
But that's as far as I've got.  I'd appreciate some help. 

A:

$z \equiv 0 \pmod 8$ is not correct because $260$ is not divisible by $8$.  Once you fix that, you will have three equations in three unknowns.   You should get $z$ directly because it has to be a single digit.   That leaves two equations in two unknowns.   The $\pmod 9$ one can only sum to $18$ or $27$.  The $\pmod {11}$ can only sum to ???  given that $x,y$ are single digits. 

A:

The divisibility test for $8$ is whether the last three digits are divisible by $8$.  So you must have $26z$ a multiple of $8$.  Hence $z=4.$
By the test for $11$ we have
$$x-y+2-6+4 \equiv 0 \pmod{11}$$
or 
$$x-y \equiv 0 \pmod{11}.$$
So we must have $x=y$.
By the test for $9$ we have 
$$x+x+2+6+4 \equiv 0 \pmod{9}$$
or
$$2x \equiv 6 \pmod{9}.$$
So $2x =6$ or $2x=15.$  So $x=3=y$.  Final answer $33264.$

