[SOLVED] Finding the sum of two unknown variables

Hello everyone,

Quote:

Given that $\displaystyle 43p + 23q = 4323$, find $\displaystyle p + q$?

The question is to find the sum of the unknown variables. Now, the variables have to be integers, and as you can see from the easy example I made, the sum could be $\displaystyle 43(100) + 23(1)=4323$ $\displaystyle \Rightarrow p+q = 101$

Of course, this is not the only solution. Try the following:

$\displaystyle 43(54) + 23(87) = 4323$

$\displaystyle \Longrightarrow p + q = 141$

I suspect that there are even more integer solutions to this equation. My method for finding the values is the old simple bruteforce "technique", where I divide 4323 by highest coefficient and that's where I get a headstart. You can of course find out the pattern, but these "cooked" equations are no good for learning.

My question is: **is there a logical process of solving for $\displaystyle p + q$?** Other than bruteforce, of course. I asked my teacher today, and he just said "that's number theory." I looked up number theory, but I was kind of lost.

These type of questions are pretty common on the SAT I tests, and I love them. My brother and I used to exchange multivariable equations like that and try to find the sum all the time.