<s>[INST] Equations with unknowns and powers $a + b +c = 17$
$a^2 + b^2 + c^2 = 101$
$a^3 + b^3 + c^3 = 623$
How does one go about solving this?
Thanks
 [/INST] Hint: 
Expand $(a + b + c)^2$ and $(a + b + c)^3$, and use the three equations you are given.
$$(a+b+c)^2 = (17)^2 = \underbrace{\color{blue}{a^2 + b^2 + c^2}}_{101} +2(\bf{ab + ac + bc})$$
$$\begin{align} (a + b+ c)^3 = (17)^3 & = \underbrace{\color{red}{a^3 + b^3 + c^3}}_{623} + 3(a^2b + a^2c + ab^2 + b^2c + ac^2 + bc^2) \\ \\&= 623 + 3(a^2b + a^2c + ab^2 + b^2c + ac^2 + bc^2 + \color{green}{abc - abc})\\ \\ & = 623 + 3\Big(\underbrace{(a + b + c)}_{17}{\bf(ab + ac + bc)}- \color{green}{\bf abc}\Big)\end{align}$$  
Spoiler: 

 Try the triple of numbers $4, 6, 7$.

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