Given $$5x+7y=c$$What is the largest value of c for which there exists exactly 3 solutions (x,y)? I've tried researching how to find the exact number of positive integer solutions for linear diophantine equations but didn't find it much help. How would I solve this type of problem?
I believe the answer is supposed to be 140.
To get the answer of 140 you must mean positive integers rather than non-negative. If you really mean non-negative the analysis would be similar but use $\geq$ and $\leq$ rather than $>$ and $<$.
A solution to $5x+7y=1$ is (-4,3) so a solution to $5x+7y=c$ is (-4c,3c) so the general solution is $(-4c+7n,-3c+5n)$. We want positive solutions so we need $-4c+7n>0$ and $-3c+5n>0$ or in other words: $$\frac{4c}{7}<n<\frac{3c}{5}$$ For the largest c such as we get only 3 solutions consider what is required to get 4 solutions: $$\frac{4c}{7}<n<n_2<n_3<n_4<\frac{3c}{5}$$ For only 4 solutions these will be sequential. I.e. $n_4=n+3$ $$\frac{4c}{7}<n<\frac{3c}{5}-3$$ So we want to find the largest $c$ which doesn't have unique integer solution to the inequality (as we don't want to get 4 solutions).
For no unique solution and maximal $c$ we want the two sides of the inequality to be consecutive integers. $$\frac{4c}{7}=\frac{3c}{5}-3-1$$ $$20c=21c-140$$ $$c=140$$