I have two natural numbers, A and B, such that A * B = AB.

Do any such numbers exist? For example, if 20 and 18 were such numbers then 20 * 18 = 2018.

From trying out a lot of different combinations, it seems as though putting the digits of the numbers together always *overestimates*, but I have not been able to prove this yet.

So, I have 3 questions:

- Does putting the digits next to each other always overestimate? (If so, please prove this.)
- If it does overestimate, is there any formula for computing by how much it will overestimate in terms of the original inputs A and B? (A proof that there's no such formula would be wonderful as well.)
- Are there any bases (not just base 10) for which there are such numbers? (Negative bases, maybe?)