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?)
Best Answer
Lets put aside the trivial answer $A=0$ and $B=0$ and consider both $A, B>0$.
You want numbers such that $A\times B = A\times10^k + B$ where $k$ is the number of digits of $B$, that is with $10^{k-1}\leq B < 10^k$. So you need $B=10^k+\dfrac{B}{A}$ with $B<10^k$. From which results $B>10^k$ and $B<10^k$.
So if there's no mistake there, the answer is no.