11,9 are co-prime to each other
2^99 mod 9 = 2^(6*16+3) mod 9 = 2³ mod 9 = 8 mod 9
2^99 mod 11 = 2(10*10-1) mod 11 = (1/2) mod 11 = 6 mod 11

From 1st condition, the remainder can be
8,17,26,35,44,53,62,71,80,89,98.
From second condition, the remainder can be
6,17,28,39,50,61,72,83,94.

The common value of both the sets is 17.

It can be found in other way using CRT
(1/9) mod 11 = 5 mod 11
(1/11) mod 9 = 1/2 mod 9 = 5 mod 9
The required answer = 8*5*11+6*5*9 mod 99
8,6 are remainders from 1st 2 steps
=(44+72) mod 99 = 116 mod 99 = 17