Problem of the Month
Hint for Problem 0: Equations in the integers

1. An exhaustive search is a reasonable approach to this problem. It can be made easier if you notice that \(x\) must be a multiple of \(8\) and that \(y\) must be a multiple of \(5\).

2. Find a positive integer \(c\) with the property that \(ax+by=c\), \(ax+by=c+1\), \(ax+by=c+2\), \(ax+by=c+3\), and \(ax+by=c+4\) all have non-negative solutions.

3. As always, it is good to work out a few small examples to try to guess a pattern. It might be useful to understand the set of all integer solutions to \(ax+by=c\) for fixed \(a\), \(b\), and \(c\) with \(\gcd(a,b)=1\). Once you do this, you might consider the integer solution \((x,y)=(u,v)\) with \(u\) negative but as close to \(0\) as possible.

4. See the hint for Question 3.

5. See the hint for Question 3.