PUMaC 2013 · 组合（B 组） · 第 2 题

PUMaC 2013 — Combinatorics (Division B) — Problem 2

[ 3 ] The number of positive integer pairs ( a, b ) that have a dividing b and b dividing 2013 can be written as 2013 n + k , where n and k are integers and 0 ≤ k < 2013. What is k ? Recall 2013 = 3 · 11 · 61.

解析

[ 3 ] The number of positive integer pairs ( a, b ) that have a dividing b and b dividing 2013 can be written as 2013 n + k , where n and k are integers and 0 ≤ k < 2013. What is k ? Recall 2013 = 3 · 11 · 61. Solution This is equivalent to choosing x , x , x , y , y , y nonnegative integers less than or 1 2 3 1 2 3 equal to 2014 so x ≤ y . There are 2015 + 2015(2014) / 2 ≡ 2 + 1 = 3 ways to choose each x , y i i i i pair, for a total of 27(2013) ways.