What is the lowest value of $m$ if $m>2$ and $m^3-3m^2+2m$ is divisible by $79$ and $83$?

The brute force method: as $m^3−3m^2+2m=m(m−1)(m−2)$, and $79,83$ are prime, you can just solve the following nine congruences: $m\equiv\alpha\pmod{79}$, $m\equiv\beta\pmod{83}$, where $\alpha,\beta\in\{0,1,2\}$. This is possible as per Chinese Remainder Theorem, and the smallest of the nine $m$'s you will get (greater than $2$) is the solution.

It is easy to solve all those congruences simultaneously: per Wikipedia, we first express $1$ as $1=79u+83v$, where $u,v$ can be found using Euclidean algorithm. In this case, as $4=83-79$ and $1=20\cdot 4-79$, we have $1=20\cdot 83-21\cdot 79$.

Now, $m\equiv\alpha\pmod{79}$ and $m\equiv\beta\pmod{83}$ resolves as $m\equiv 20\cdot 83\alpha-21\cdot 79\beta\pmod{79\cdot 83}$, i.e. $m\equiv 1660\alpha-1659\beta\pmod{6557}$. This gives us the following table:

$$\begin{array}{r|r|r|r}\alpha&\beta&m\pmod{6557}&\text{smallest }m\gt 2\\\hline0&0&0&6557\\0&1&4898&4898\\0&2&3239&3239\\1&0&1660&1660\\1&1&1&6558\\1&2&4899&4899\\2&0&3320&3320\\2&1&1661&1661\\2&2&2&6559\end{array}$$

so the smallest solution seems to be $m=1660$.