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HMMT 十一月 2023 · 冲刺赛 · 第 9 题

HMMT November 2023 — Guts Round — Problem 9

专题
Discrete Math / 离散数学
难度
L3
来源
HMMT

题目详情

  1. [7] The largest prime factor of 101 101 101 101 is a four-digit number N . Compute N . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . HMMT November 2023, November 11, 2023 — GUTS ROUND Organization Team Team ID#
解析
  1. [7] The largest prime factor of 101 101 101 101 is a four-digit number N . Compute N . Proposed by: Arul Kolla Answer: 9901 Solution: Note that 101 101 101 101 = 101 · 1 001 001 001 = 101 · 1001 · 1 000 001 3 = 101 · 1001 · (100 + 1) 2 = 101 · 1001 · (100 + 1)(100 − 100 + 1) = 101 · 1001 · 101 · 9901 2 = 101 · 1001 · 9901 2 = (7 · 11 · 13) · 101 · 9901 , and since we are given that the largest prime factor must be four-digit, it must be 9901. One can also check manually that it is prime.