<s>[INST] Knowing that m and n are two positive integers, solve the equation m/n = n.m (ex.: 5/2 = 2.5) Knowing that $m$ and $n$ are two positive integers, find all the solutions for the equation: 
$\frac{m}{n} = n.m$ 
For example, the pair $(m=5, n=2)$ is a solution because $\frac{5}{2} = 2.5$ while $(m=294, n=17)$ comes close to a solution but it is not because $\frac{294}{17} = 17.29411...$ .
 [/INST] $m/n = n.m= n + m/10^k$ so $10^kn^2 + mn - 10^km$
So by quadratic equation $n = \frac{-m + \sqrt{m^2 + 4m*10^{2k}}}{2*10^k} $
Notice:
m has k digits. 
So n = $\frac{-m + \sqrt{m^2 + 4m*10^{2k}}}{2*10^k}$ has  $\lceil k/2 \rceil$ digits.  But m/n $\approx$ n then has   ${\lceil k/2 \rceil}$ digits, so $m$ has between $\lfloor k/2 \rfloor$ and ${\lceil k/2 \rceil}$ digits. But m has $k $ digits.  So k = 1 or 2  and m has one or 2 digits and n has 1 digit.  $nm$ is a multiple of $10^k$ but m/n has only 1 digit so k = 1 and m has 1 digit.
$n = \frac{-m + \sqrt{m^2 + 400}}{20} $
So $m^2 + 400$  is a perfect square.  Only possibility  is m =5 so n = 2.  
</s>