HMMT 十一月 2025 · 冲刺赛 · 第 32 题
HMMT November 2025 — Guts Round — Problem 32
题目详情
- [17] Compute the smallest positive integer n for which n (written in base 10) ends in 123.
解析
- [17] Compute the smallest positive integer n for which n (written in base 10) ends in 123. Proposed by: Derek Liu Answer: 867 n Solution: Note that n ≡ 3 mod 8, so n ≡ 3 mod 8. Thus, 3 n n ≡ n ≡ 3 mod 5 , so n ≡ 27 mod 40. Then, 7 n n ≡ n ≡ − 2 mod 25 . 21 Cubing both sides and using the fact that n ≡ n mod 25, 3 n ≡ ( − 2) = − 8 mod 25 , so n ≡ 67 mod 200. Then, 67 n n ≡ n ≡ − 2 mod 25 . 201 Cubing both sides and using the fact that n ≡ n mod 125, 3 n ≡ ( − 2) = − 8 mod 125 , so n ≡ 867 mod 1000.