[Math] Are there any two numbers such that multiplying them together is the same as putting their digits next to each other

natural numbers

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:

  1. Does putting the digits next to each other always overestimate? (If so, please prove this.)
  2. 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.)
  3. Are there any bases (not just base 10) for which there are such numbers? (Negative bases, maybe?)

Best Answer

I have two natural numbers, $A$ and $B$, such that $A \times B = AB$.

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

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.

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