Before considering reductions modulo particular numbers, let's observe that A and B must be relatively prime, because if A and B had any common factor F, then reducing both sides mod F would give N = 1 mod F and N = 2 mod F, so that 1 = 2 mod F, which is only possible for F=1, a case that is excluded from factorizations anyway.

Consider mod 2. If A is even, then 37861(A) = 1 mod 2, and if A is odd, then 37861(A) = 1 mod 2 again. So the actual number represented by the digit'strings is odd. Now this forces B to be odd, for if B were even, then 42050522(B) = 0 mod 2. So we have established that the number is odd and B is odd.

Consider mod 3. To simplify things, let's reduce each digit mod 3, giving the numerals as 01201(A) and 12020222(B). Now look at the possibilities:
A = 0 mod 3 ---> N = 1 mod 3 ... B = 0 mod 3 ---> N = 2 mod 3
A = 1 mod 3 ---> N = 1 mod 3 ... B = 1 mod 3 ---> N = 2 mod 3
A = 2 mod 3 ---> N = 2 mod 3 ... B = 2 mod 3 ---> N = 2 mod 3
This proves that N = 2 mod 3 (no choice about that) and A = 2 mod 3 (because the mod 3 reduction of N must give the same answer independent of the base used in its expression).

At this point, we know that A = 2 mod 3, B = 1 mod 2, and N = 5 mod 6.

Consider mod 4. The numerals reduce to 33021(A) and 02010322(B), and the possibilities are:
A = 0 mod 4 ---> N = 1 mod 4
A = 1 mod 4 ---> N = 1 mod 4 ... B = 1 mod 4 ---> N = 2 mod 4
A = 2 mod 4 ---> N = 1 mod 4
A = 3 mod 4 ---> N = 3 mod 4 ... B = 3 mod 4 ---> N = 2 mod 4
Here we run into trouble! There is no compatible choice, therefore, there CANNOT be any two bases A and B whatsoever, which will represent the same number N by these two particular strings of digits.

Either I have made a calculation error (but I've rechecked, and I don't see one), or the original problem contains an error. Perhaps it is a typo in one of the digits? Perhaps instead it is a misinterpretation of what the number conversion operation actually did. Is it certain that it was simply a base conversion from one unspecified base to another?

Hope this helps!

--Stuart Anderson