From reading the problem if we had just one more tourist, i.e. we would be able to evenly divide them among the 22 seats so . The other equation says that if we had one bus less then the tourists can be evenly divided with seats in each both, so, .
Thus, we have two equations,
The last equation tells us that because the LHS is positive while RHS would not be.
Note that is an integer so the right hand side must be an integer.
Since by the conditions of the problem .
And the contrainst that we found we need to find all with such that is an integer.
We can throw away half the cases if we realize that the denominator and numerator have opposite parity. And so the denominator cannot be even. That happen only when is even. Thus, we need to check the cases by hand which is fast.
But I notice (without doing that) a trivial answer of for the makes the denominator 1.
Note: There might be other solutions which I did not check for.