Short Setup; Copious Constraints

Relevant wiki: Number Base - Problem Solving

Let $x$ , $y$ and $z$ denote the first, second and third digits of $N$ in base 9 so that:

$81x+9y+z=49z+7y+x$ or $y=8(3z-5x)$ .

Since $0\le y<7$ (it appears as a digit in base 7), the integer $n=3z-5x$ is zero (otherwise 8n would be greater than 7).

Hence $y$ , the middle digit, is zero. Moreover, $0<z<7$ (since N has three digits in base 7); and since $3z = 5x$ , we have $z=5$ and $x=3$ .

Therefore, we have $N={ 305 }_{ 9 }={ 503 }_{ 7 }={ \boxed { 248 } }_{ 10 }$ .