Problem 5 · Use a shared root and factor pairs · 4 points
Made by AI and not reviewed by a real person.
The equations x² − ax + 24 = 0 and x² − bx + 36 = 0 each have two distinct positive integer solutions, and they share one solution. What is the smallest possible value of a + b?
- 23
- 22
- 26
- 34
- 29
Show a hint
- Think about how the constant term limits the positive integer solutions of each equation. Which numbers could be a solution to both?
- Call the shared solution r. Find the other solution of each equation in terms of r, then check the possible values of r. Remember that each equation needs two distinct solutions.
Show the answer and solution
Answer: A — 23
The shared solution r must divide both 24 and 36, so r is 1, 2, 3, 4, 6, or 12. The other solutions are 24/r and 36/r, giving a + b = 2r + 60/r. For r = 1, 2, 3, 4, and 12, the totals are 62, 34, 26, 23, and 29. The tempting total 22 comes from r = 6, but then the second equation has the repeated solution 6. Thus the minimum is 23, achieved when the solutions are 4 and 6 for one equation, and 4 and 9 for the other.