<s>[INST] Find integer solution to the given system of equation. find all  positive integers $a,b,c$ such that
$abc=24$
$ab+bc+ca=38$
If particular values are given then we can easily find the solution but I am searching for some short general method.  Is there any sufficient condition for the constant values to have integer solution?
 [/INST] $ab+bc+ca=38 \Rightarrow a^2(b+c)+24=38a$. From the function $f(a)=a^2(b+c)-38a+24$ we get $38^2 \ge 4\cdot 24\cdot (b+c) \Rightarrow 15 \ge (b+c)$. You have now 13 values for b+c but not for every value will $38^2-4\cdot 24\cdot (b+c)$ be a perfect square and we also need $a_{1,2}=\frac{38 \pm \sqrt{38^2-4\cdot 24\cdot (b+c)}}{2(b+c)}$ to be an integer. We always get a and (b+c), but b and c can be calculated using $bc=\frac{24}a$. You will get (1,2,12) and his permutations as the only solution.
Note: we see from the solution that $a+b+c=15$ is always constant, so proving that would give us an even more simple solution bcs then we know that a,b,c are the solutions to the equasion $x^3-15x^2+38x-24=(x-1)(x-2)(x-12)=0$
This is as much as I know, if there is a faster solution that can covert them all please let me know.
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