## KingGeorge 4 years ago [UNSOVED] What is the largest number (in terms of a, b, c) that can not be expressed in the form $ax+by+cz$ where $x,y,z≥0$$gcd(a,b,c)=1$and all values are non-negative integers? As an example, take $6x+10y+15z$Here, the largest integer that can't be expressed in this form given $x, y, z \geq 0$is 29. If instead you're solving the same problem, only with $ax+by$instead of$ax+by+cz$ with the same conditions that every value is a non-negative integer, the largest integer not expressible in this form is given by the formula$ab-a-b$

1. anonymous

IMO problem?

2. anonymous

A similar problem appear in IMO 1983, 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 that cannot be expressed in the form $$xab+yca+zab$$, where$$x, y,$$ and $$z$$ are non-negative integers.

3. KingGeorge

no, the very last part of a homework problem from our textbook. I've figured out the two variable case without too much trouble, but for some reason, I just don't know how to generalize it.

4. KingGeorge

Well, that appears to be almost the generalization I'm looking for...

5. anonymous

Yes, I think we still need to do some work, btw what level of mathematics is this?

6. anonymous

Like high-school , undergrad, etc...

7. KingGeorge

undergrad - intro to number theory

8. anonymous

Cool, I thought so.

9. KingGeorge

I've worked through a few other cases while varying c. So far I've gotten 25 if a=6, b=10, c=11. And 23 if a=6, b=10, c=9.

10. KingGeorge

Here a few more cases 11 if (a, b, c) = (4, 5, 6) also 11 if (a, b, c) = (4, 5, 7) also 11 if (a, b, c) = (4, 6, 9)

11. KingGeorge

My best guess currently, for those of you looking at this, is that the solution is the smallest integer that satisfies these equations$\begin{matrix} ab-a-b-d = cx \\ ac-a-c-d=by \\ bc-b-c-d=az \end{matrix}$where d is the solution we're looking for, and x, y, z are still all positive.