PUMaC 2015 · 代数（A 组） · 第 1 题

PUMaC 2015 — Algebra (Division A) — Problem 1

[ 3 ] How many integer pairs ( a, b ) with 1 < a, b ≤ 2015 are there such that log b is an integer? a 2 3 2

解析

[ 3 ] How many integer pairs ( a, b ) with 1 < a, b ≤ 2015 are there such that log b is an integer? a Solution: We do casework on how large log b is. Note that log b > 0 since b > 1 and a ≥ 2 while a a b ≤ 2015 so log b ≤ log 2015 < 11. a 2 If log b = 1, then a = b so there are 2014 such pairs. a 2 If log b = 2, then a = b ≤ 2015 so a < 45 so there are 43 such pairs. a 3 If log b = 3, then a = b ≤ 2015 so a < 13 so there are 11 such pairs. Note that we can find a 3 3 this bound easily by 12 = 144 · 12 < 150 · 12 = 1800 and 13 = 169 · 13 > 160 · 13 = 2080. We continue for each integer value of log b less than 11 and in total there are 2014 + 43 + 11 + a 5 + 3 + 2 + 1 + 1 + 1 + 1 = 2082 such pairs. Author: Roy Zhao 2 3 2