The question is as stated in the title. I feel like the question is geared towards some kind of case that yields to the divisibility test for $9$, but I think an argument can be made for other numbers as well. For example in the sequence $100$ ,...,$117$ we have of course $3 \mid 102$ and $9 \mid 117$, but we also have $2 \mid 110$ and $4 \mid 112$. And secondly in sequences where the digit sum is above $9$, we still have divisors that are multiples of $9$. For example $18 \mid 990$.
So if anyone could help me with a proof, I'd be really grateful. Thank you for your help.