Tell Me The Number Of Digits First!

Let us first establish that the answer is an integer. Let $N$ be the answer to this question. Since the number of digits of $N$ is an integer (call it $M$ ), then 111 times this $M$ is also an integer. And because $111M$ is an integer, then so does the answer in question, $N$ .

The number of digits of an integer $n$ is $\lfloor \log_{10} n \rfloor + 1 = \lfloor \log_{10} n+ 1 \rfloor$ . This result can be shown in the article: finding the number of digits .

So we have $M = \lfloor \log_{10} N+ 1 \rfloor$ . And we want to solve the equation $111M = N$ , or equivalently $111 \lfloor \log_{10} N+ 1 \rfloor = N$ .

Trial and error shows that $\lfloor \log_{10} N + 1 \rfloor = 3$ and $N = 333$ .

Now we want to prove that the answer is unique. Suppose that there is there exists a solution for $P \ne 333$ that is also the answer to this question.

If $P <333$ , then $\lfloor \log_{10} P + 1 \rfloor = 1,2,3 \Rightarrow 111 \lfloor \log_{10} P + 1 \rfloor = 111,222$ . But neither of these solutions satisfy the condition.

If $333<P<1000$ , then $\lfloor \log_{10} P + 1 \rfloor = 3 \Rightarrow 111 \lfloor \log_{10} P + 1 \rfloor = 333$ . But this contradicts our condition as well.

If $P \geq 10^4$ , then tabulating the values of $\lfloor \log_{10} + 1 \rfloor$ versus the range of $P$ gives

$\begin{array} { | c | c | c | c | c | c | } \hline \lfloor \log_{10} P+ 1 \rfloor & 4 & 5 & 6& 7 & \cdots \\ \hline 111 \lfloor \log_{10}P + 1 \rfloor & 444 & 555 & 666& 777 & \cdots \\ \hline \text{range}(P) & [10^4,10^5) & [10^5,10^6) & [10^6,10^7)& [10^7,10^8) & \cdots \\ \hline \end{array}$

Note that the values of $111 \lfloor \log_{10}P + 1\rfloor$ grows linearly, whereas the smallest value of $P$ grows exponentially. So it is impossible for the value of $111 \lfloor \log_{10}P + 1\rfloor$ to catch up to even the smallest corresponding value of $P$ . In other words, the difference between these two values grows larger and larger. Hence, there can't be any solution for $P\geq 10^4$ as well. Hence, $N = \boxed{333}$ is the only answer to this question.

3 digits in answer so 111*3=333