PUMaC 2015 · 团队赛 · 第 3 题

PUMaC 2015 — Team Round — Problem 3

[ 3 ] How many ordered pairs ( a, b ) of positive integers with 1 ≤ a, b ≤ 10 are there such that in the geometric sequence whose first term is a and whose second term is b , the third term is an integer?

解析

[ 3 ] How many ordered pairs ( a, b ) of positive integers with 1 ≤ a, b ≤ 10 are there such that in the geometric sequence whose first term is a and whose second term is b , the third term is an integer? 2 b Solution: The next term in the sequence is and hence given a choice of b , we will count a 2 how many choices of 1 ≤ a ≤ 10 are there such that a | b . Doing so, we get the pairs: (1, 1), (1, 2), (2, 2), (4, 2), (1, 3), (3, 3), (9, 3), (1, 4), (2, 4), (4, 4), (8, 4), (1, 5), (5, 5), (1, 6) (2, 6), (3, 6), (4, 6), (6, 6), (9, 6), (1, 7), (7, 7), (1, 8), (2, 8), (4, 8), (8, 8), (1, 9), (3, 9), (9, 9) (1, 10), (2, 10), (4, 10), (5, 10), (10, 10) So there are 33 such choices. Author: Roy Zhao