HMMT 十一月 2023 · 冲刺赛 · 第 24 题
HMMT November 2023 — Guts Round — Problem 24
题目详情
- [12] Compute the smallest positive integer k such that 49 divides . k . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . HMMT November 2023, November 11, 2023 — GUTS ROUND Organization Team Team ID#
解析
- [12] Compute the smallest positive integer k such that 49 divides . k Proposed by: Edward Yu Answer: 25 2 k a Solution: The largest a such that 7 | is equal to the number of carries when you add k + k in k base 7, by Kummer’s Theorem. Thus, we need two carries, so 2 k must have at least 3 digits in base