September 2025
Suppose \(a\), \(b\), and \(c\) are positive integers. In this problem, a non-negative solution to the equation \(ax+by=c\) is a pair \((x,y)=(u,v)\) of integers with \(u\geq 0\) and \(v\geq 0\) satisfying \(au+bv=c\). For example, \((x,y)=(7,0)\) and \((x,y)=(3,3)\) are non-negative solutions to \(3x+4y=21\), but \((x,y)=(-1,6)\) is not.
Determine all non-negative solutions to \(5x+8y=120\).
Determine the largest positive integer \(c\) with the property that there is no non-negative solution to \(5x+8y=c\).
In Questions 3, 4, and 5, \(a\) and \(b\) are assumed to be positive integers satisfying \(\gcd(a,b)=1\).
Determine the largest non-negative integer \(c\) with the property that there is no non-negative solution to \(ax+by=c\). The value of \(c\) should be expressed in terms of \(a\) and \(b\).
Determine the number of non-negative integers \(c\) for which there are exactly \(2025\) non-negative solutions to \(ax+by=c\). As with Question 3, the answer should be expressed in terms of \(a\) and \(b\).
Suppose \(n\geq 1\) is an integer. Determine the sum of all non-negative integers \(c\) for which there are exactly \(n\) nonnegative solutions to \(ax+by=c\). The answer should be expressed in terms of \(a\), \(b\), and \(n\).
Fact: You may find it useful that for integers \(a\) and \(b\) with \(\gcd(a,b)=1\), there always exist integers \(x\) and \(y\) such that \(ax+by=1\), though \(x\) and \(y\) may not be non-negative.