HMMT 十一月 2024 · 冲刺赛 · 第 9 题

HMMT November 2024 — Guts Round — Problem 9

[7] Compute the remainder when 1 002 003 004 005 006 007 008 009 is divided by 13 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . HMMT November 2024, November 09, 2024 — GUTS ROUND Organization Team Team ID# 12 9 5

解析

[7] Compute the remainder when 1 002 003 004 005 006 007 008 009 is divided by 13. Proposed by: Pitchayut Saengrungkongka Answer: 5 Solution: Note that 13 | 1001. Thus we can repeatedly subtract any multiple of 1001 from this number without changing the remainder. In particular, we can repeatedly subtract multiples of 1001 from the left to the right, as follows. 1 002 003 004 005 006 007 008 009 −→ 1 003 004 005 006 007 008 009 −→ 2 004 005 006 007 008 009 −→ 2 005 006 007 008 009 −→ 3 006 007 008 009 −→ 3 007 008 009 −→ 4 008 009 −→ 4 009 −→ 5 . 12 9 5