In 2002, a math competition of India named Pradanya meant to ask two 6-digit numbers $P=abcdef$ and $Q=defabc$ such that $P=6 Q$. We can write $P=1000X+Y, Q=1000Y+X$, where $X=abc$ and $Y=def$ are 3-digit numbers. Then we get $$\frac{X}{Y}=\frac{6000-1}{1000-6}=\frac{857}{142}.$$ Hence $P=857142.$
This interesting number gives rise to the following question:
Find $2n$-digit numbers $P_{2n}$ and $Q_{2n}$ are such that $P_{2n} = m Q_{2n}$, $m \in \{2,3,...,9\}$, where $P_{2n} = \{X_n\} \{Y_n\}, ~ Q_{2n}=\{Y_n\} \{X_n\}; \{X_n\} = x_1x_2x_3...x_n,~ \{Y_n\}=y_1y_2y_3...y_n$ are $n$-digit numbers. We will get the equation $$\frac{(m 10^n-1)}{{10^n-m}}=\frac{\{X_n\}}{\{Y_n\}}$$ to solve, Numerics suggest no solution for $n=1,2,4,5$ and one solution $m=6$ for $n=3$ which has already produced $P_6=857142.$ The question is: Are there other such numbers as $P_6$ and what could be possible mathematical connections?