Find a 13 digit positive integer N whose base ten representation consists entirely of 8s and 9s
such that 213 divides N.
(In reply to re: answers- spoiler
I was short of time to write down my "induction reasoning "and your presentation provided a faultless proof, thus complementing my layout.
My final answer was reached by adding 8 or 9 as a 1st digit followed by a previous result i.e. 8,88,888,9888 etc (thanks for correcting my typo - I've edited my post ).
All said , it was a very nice problem!