How to calculate a missing middle digit of a multiplication between 2 big numbers using modular arithmetic? What elementary number theory methods can I use for solving this type of questions?
One of these kind of problems is to find $x$ where $985242x6565 = 172195\cdot 572167$ without multiplying the numbers again.
I tried doing, let's call $985242x6565$ $N$, $N\text{mod}10^5$ to somehow get to the $5th$ digit but this is doesn't get me anywhere and in hindsight it doesn't make any sense. How should I proceed?

The remainder of the division by 11 is equal to the alternating sum of the digits of the number. Because
$1 \equiv 1$ (mod 11)
$10 \equiv -1$ (mod 11)
$100 \equiv 1$ (mod 11)
$1000 \equiv -1$ (mod 11)
$10000 \equiv 1$ (mod 11)
and so on...
Thus,
$172195 \equiv (-1+7-2+1-9+5) \equiv 1$ (mod 11)
$572167 \equiv (-5+7-2+1-6+7) \equiv 2$ (mod 11)
$172195⋅572167 \equiv 1\cdot2 \equiv 2$ (mod 11)
$985242x6565 \equiv (9-8+5-2+4-2+x-6+5-6+5) \equiv 4+x$ (mod 11)
and $x\equiv9$ (mod 11).
Answer: $x=9$
