<s>[INST] $a+b+c+ab+bc+ca+abc=1000$ find the minimum value of $a+b+c$ Given $a+b+c+ab+bc+ca+abc=1000$.
Find the minimum value of $a+b+c$.
Now we are considering $a$, $b$, $c$ to be integers and here in lies the pertinent problem as I could not get an answer in integers but in fractions.
 [/INST] Trial and error can be avoided.  Note that minimizing $x+y+z$ where the positive real numbers $x,y,z$ have a specified product $1001$ is achieved by making $x,y,z$ as nearly equal as possible.  
Here we minimize $(1+a)+(1+b)+(1+c)$ when we minimize $a+b+c$.  Since $a,b,c$ are required to be (positive?) integers, the most equal we can make the factors is by assigning the primes $7,11,13$ to $(1+a),(1+b),(1+c)$ in some order.  Although this assignment is not unique, the minimum sum $a+b+c = 6+10+12$ will be unique.
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