I think some progress can be made by looking at the divisbility properties of the bases. Since the two digit patters represent the same actual number N, they must be equivalent modulo anything you choose. Let the base of 37861 be A and the base of 42050822 be B.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