Problem 7 · Count guaranteed prime factors · 5 points
Made by AI and not reviewed by a real person.
Write down any nine consecutive positive whole numbers, cross out one, and multiply the other eight. Which number is guaranteed to divide the product?
- 288
- 720
- 144
- 432
- 1008
Show a hint
- Think about which small prime factors must survive, even if you cross out a number that supplies several of them.
- In a group of nine consecutive numbers, count the evens, the multiples of 4, and the multiples of 3. How many of each can one crossed-out number take away?
Show the answer and solution
Answer: C — 144
Nine consecutive numbers contain at least four evens, at least two multiples of 4, and exactly three multiples of 3. Crossing out one leaves enough factors for at least 2⁴ × 3² = 144. The trap is assuming an extra factor must survive too: starting with 1 through 9, crossing out 8, 9, 5, or 7 gives 144 × 315, 144 × 280, 144 × 504, or 144 × 360, respectively—ruling out 288, 432, 720, and 1008.