Alex's Simple Division

The value of $n = 11$

So, the largest possible divisor of:

$2619941878888....88888626894701 (8$ written $1993$ times $) = 11^{11} = 285,311,670,611$

So, the answer is $\boxed{11}$ .

java code:

public class brilliant201409091555{
    static int count = 0;
    public static void main(String args[]){
        String number = "261994187";
        for(int i=0;i<1993;i++) number = number + "8";
        number = number + "626894701";
        process(number);
        System.out.println(count);
    }
    private static void process(String number){
        String quotient = "";
        int remainder = 0;
        for(int i=0;i<number.length();i++){
            remainder *= 10;
            remainder += (number.charAt(i)-48);
            if(remainder<11){
                quotient += "0";
                continue;
            }
            quotient += remainder/11;
            remainder %= 11;
        }
        if(remainder==0){
            count++;
            process(quotient);
        }
    }
}

output:

11
Press any key to continue . . .

Note: I used the concept of BigInteger by representing integers with strings.