1. The number 59 can be made by adding non-negative integral multiples of A and B in exactly one way.

2. Some numbers below 59 cannot be produced in this way at all, while some can be produced in one way and others in more than one way as sums of non-negative integral multiples of A and B.

3. All numbers above 59 can be produced in more than one way as sums of non-negative integral multiples of A and B.

What two numbers can A and B be?

You claim:    "Every number from 0 to AB-1 can be dome in some number of ways. 

Not so: Given 4:9 as the integers in puzzle, the 1st numbers expressable are : 4,8,9,13,17,18 etc

That was the reason for condition 2 (some numers not being expressable) to void any solution naming  a number 1 as one of the integers

N Every number from AB and above can be done on one more way than itself minus AB.