The Conductor of Two Naturals is the largest number which cannot be written as mb+nc

This paper presents a short but non-obvious and interesting theorem in Number Theory that I originally discovered while working on a problem. This theorem states that \( bc - b - c \) is the largest number which \emph{cannot} be written as \( mb + nc \). Given all \( b, c, m and n \in \mathbb{N} \) . In this article I prove the above statement and also show a problem where this theorem could be directly applied to considerably make the problem easier.