1. **State the problem:** We are given the equation $x(r - 7) + 3 = 17x + 25$ where $r$ is a positive integer. We want to find the values of $r$ for which the equation has exactly one solution for $x$, and specifically identify which value of $r$ cannot produce exactly one solution. 2. **Rewrite the equation:** $$x(r - 7) + 3 = 17x + 25$$ Bring all terms involving $x$ to one side and constants to the other: $$x(r - 7) - 17x = 25 - 3$$ $$x[(r - 7) - 17] = 22$$ $$x(r - 24) = 22$$ 3. **Analyze the solution:** - If $r - 24 \neq 0$, then $$x = \frac{22}{r - 24}$$ which is a unique solution. - If $r - 24 = 0$, i.e., $r = 24$, then the equation becomes: $$x \cdot 0 = 22$$ which is $$0 = 22$$ This is a contradiction, so no solution exists. 4. **Conclusion:** - For all $r \neq 24$, there is exactly one solution. - For $r = 24$, there is no solution. Since the problem asks for the value of $r$ that **cannot** produce exactly one solution, the answer is: $$\boxed{24}$$