Uncle Mané has a broken calculator that doesn't have the 0 key, and the display never shows 0 after any operation. For example, if he multiplies \(3 \times 67\), he gets 21 instead of 201.

Uncle Mané multiplied two two-digit numbers on his calculator and got the number 11 on the display. What are the possible numbers he multiplied?
As the calculator does not have the key $O$, the product of two two-digit numbers on this calculator is greater than or equal to $11 \times 11=121$ and less than $100 \times 100=10000$, the possible answers for the product are: 1001, 1010, and 1100. For each of the cases, we have:

- $1001=11 \times 91=13 \times 77$, two possible solutions;
- $1010=101 \times 10$ and since 101 is prime, there is no solution in this case;
- $1100=11 \times 2^{2} \times 5^{2}=25 \times 44$ is the only solution since neither of the two factors can be simultaneously divisible by 2 and 5.

Therefore, the possible products performed by Uncle Mané are $11 \times 91$ or $13 \times 77$ or $25 \times 44$.