$a+b+c+ab+bc+ca+abc=1000$ 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.
 A: Without loss of generality, suppose that $a\leq b\leq c$.  Also, write $N:=1001$.  Since $$(a+1)(b+1)(c+1)=N\,,$$ we have that $$|a+1|\,|b+1|\leq N\,.$$  This clearly means
$$|a+1|+|b+1|\leq N+1\,.$$
(If $x$ and $y$ are positive integers with $xy\leq N$, then $(x-1)(y-1)\geq 0$, which implies that $x+y\leq xy+1\leq N+1$.)
Thus,
$$-(a+1)-(b+1)\leq |a+1|+|b+1|\leq N+1\text{ or }a+b\geq -N-3\,.$$
Since $c+1$ is clearly positive, we have $c+1\geq 1$, which means $c\geq 0$.  That is,
$$a+b+c\geq -N-3\,.$$
The equality holds iff $(a,b,c)=(-N-1,-2,0)$.

If $a$, $b$, and $c$ are required to be nonnegative, then by the AM-GM Inequality, we have 
$$\frac{(a+1)+(b+1)+(c+1)}{3}\geq \sqrt[3]{(a+1)(b+1)(c+1)}=\sqrt[3]{1001}>10\,.$$
That is, $a+b+c>27$, or $a+b+c\geq 28$.  As $(a,b,c)=(6,10,12)$ works, the minimum value of $a+b+c$ is indeed $28$.  (Unlike the first part of this answer, $1001$ can't be replaced by an arbitrary positive integer $N$.  For each $N$, a different analysis is required.)
A: Hint:
Expand $$(a+1)(b+1)(c+1)$$ 
And notice $$1001=7\cdot11\cdot13$$
A: 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.
A: I am assuming here $a=b=c$ since $1000$ is the cube of $10$.
$$(a+1)(b+1)(c+1) = 1000$$
now, $(a+1)^3 = 1000$;
$$a+1 = 10\iff
a=9$$
so, $a+b+c = 3*9 = 27$. 
Just that we can get the integer value but I cannot say $27$ is the minimum value for $a+b+c$.
