<s>[INST] How many numbers less than $1000$ with digit sum to $11$ and divisible by $11$ How many positive (integers) numbers less than $1000$ with digit sum to $11$ and divisible by $11$?
There are $\lfloor 1000/11 \rfloor = 90$ numbers less than $1000$ divisible by $11$.
$N = 100a + 10b + c$ where $a + b + c  = 11$ and $0 \le a, b, c \le 9$
I got $\binom{13}{2} - 9 = 69$ solutions.
 [/INST] $N=a+10b+100c$. $a-b+c=0$ or $a-b+c=11$. Also we have $a+b+c=11$. We get two cases:


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*$a-b+c=0,\;a+b+c=11$ from which we get $2a+2b=11$; impossible.

*$a-b+c=11,\;a+b+c=11$ from which we get $a+c=11,\; b=0$. They are $209, 308, 407, 506, 605, 704, 803, 902$.

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