The questions goes as follows:
Let $a$ , $b$ and $c$ be positive integers, no two of which have a common divisor greater than $1$. Show that $2abc - ab - bc - ca$ is the largest integer which cannot be expressed in the form $xbc + yca + zab$, where $x, y, z$ are non-negative integers.
My solution is something like this:
If $2abc - ab - bc - ca$ is expressible in the given form, then $2abc$ must be expressible in the same form $xbc + yca + zab$, but instead where $x, y, z$ are positive integers. Since a,b,c are arbitrary numbers which specify some conditions, assuming $a>b>c$ does not cause any loss in generality (or should it?) . So, now $2abc=c \times ab+b \times ca$. We must break $c$ and $b$ into smaller numbers to express $2abc$ in the required form, or in other words, we must have the relation $z_1 \times bc=x_1 \times ab+y_1 \times ca$ for some $x_1,y_1,z_1 \in N$. But both $ab,ac$ are greater than $bc$. Hence no such representation of $2abc$ exists. And since no such representations exist, $2abc - ab - bc - ca$ cannot be expressed in the form $xbc + yca + zab$, where $x, y, z$ are non-negative integers (because then it would imply $2abc$ be expressed in some form of our own,i.e., with natural numbers).
So is my solution correct? If not, then where does the fault lie in this proof? And please clear the doubt above (italicised above).