You are a chinese general and you want to count your army. Your estimate is 790,000 - 810,000. Propose the counting to determine the result unambiguously.
The soldiers can only count from to 1 to 12.
As an example remainders for the task, this information is given: every even counting of the soldiers left the remainder of $0$ and odd counting left the remainder of $1$. However, @Sandeep Silwal suggested, that this is wrong and it could be the other way around.
I tried both. In Sage software I got the following results:
CRT_list([1,1,1,0], [5,7,11,12]) 4236 CRT_list([0,0,0,1], [5,7,11,12]) 385
The first query is for even counting -> remainder $0$, odd counting -> remainder $1$.
The second one is the opposite.
Neither of the results is in the range $\langle 790000, 810000 \rangle$. What's more, according to the theorem, the number of the army should be $\le n_1 \cdot n_2 \cdot n_3 \cdot \dots$, where $n_i$ is the number in the modulo operation (am I right? I found this information in a few places). If the soldiers can count up to twelve, there is no possibility to find any coprime numbers that would multiply to give a result big enough for the range $\langle 790000, 810000 \rangle$.
How to approach this problem?