Linear congruence problem

I have a linear congruence problem that I am hoping can be solved. In short, I have a large number of linear congruences and would like a way to know which will generate the smallest solution, without having to do too many calculations.

Here is an example:

One equation from each set is combined with one from each other set, therefore a large number of possibilities. I would like to know which of those possibilities will give the smallest solution.

3a+1 or 3b+2

5b+0 or 5b+1 or 5b+4

7c+2 or 7c+3 or 7c+4 or 7c+5

11d+1 or 11d+3 or 11d+5 or 11d+6 or 11d+8 or 11d+10

Combining the equations would give the following general equation: 1155x+z, where z is different in each case. I would like to know which possibility gives the smallest z. A method that explores every possibility would not be feasible if there were a few hundred sets of equations like the above.

I am hoping someone will have some idea whether it would be possible to find some easy way to do this.