HMMT 十一月 2025 · 冲刺赛 · 第 1 题
HMMT November 2025 — Guts Round — Problem 1
题目详情
- [5] Compute the remainder when 1000! is divided by 1001. □ □ □
解析
- Clearly no 4-digit square has the same digit repeated 4 times. If there exists such a 5-digit square, either its last 2 digits are the same, or its first 3 digits are the same. (In either case, this digit is nonzero.) We consider the former case first. Observe no square ends in 22, 33, 77, or 88 because no square ends in 2, 3, 7, or 8. Furthermore, no square ends in 11, 55, 66, or 99 because no square is 2 or 3 modulo 4. Thus, such a square would have to end in 44. No square n ends in 4444, as otherwise n/ 4 would be a square ending in 11. Thus, we need only check squares of the form 4 ab 44 where either a or b is 4. It is easy to check that 44944 is the only such square. For the latter case, it suffices to check squares starting with 111, 222, 333, and 444. The only such square is 22201, which does not satisfy the required property. © 2025 HMMT