The example you've given has several different solutions for the two numbers.
Starting from the HCF being 11, let's say one number is 11 * A, and the other is 11 * B, where A and B have NO common factor (because if they did, the HCF would be more than 11, okay?)
So their LCM is 11 * A * B which is given as 7700, therefore A * B = 700.
Now factoring 700 into "co-prime" factors gives us 4 * 7 * 25. You can see that if you split up the 4 between A and B, or the 25, you'd break the condition that A and B are to have no common factor. So the possible answers for A and B are (1) 25 and 28, (2) 7 and 100, (3) 4 and 225, or (4) 1 and 700.
Multiplying by 11, the possible answers to the original question "What two numbers have an HCF of 11 and an LCM of 7700?" are (1) 275 and 308, (2) 77 and 1100, (3) 44 and 2475, or (4) 11 and 7700. The more factors there are in the LCM, the more possible pairs of numbers there would be with the given HCF and LCM.