HMMT 十一月 2018 · GEN 赛 · 第 7 题

HMMT November 2018 — GEN Round — Problem 7

Anders is solving a math problem, and he encounters the expression 15!. He attempts to simplify √ this radical by expressing it as a b where a and b are positive integers. The sum of all possible distinct values of ab can be expressed in the form q · 15! for some rational number q . Find q .

解析

Anders is solving a math problem, and he encounters the expression 15!. He attempts to simplify √ this radical by expressing it as a b where a and b are positive integers. The sum of all possible distinct values of ab can be expressed in the form q · 15! for some rational number q . Find q . Proposed by: Nikhil Reddy Answer: 4 11 6 3 2 1 1 5 3 1 1 Note that 15! = 2 · 3 · 5 · 7 · 11 · 13 . The possible a are thus precisely the factors of 2 · 3 · 5 · 7 = ab ab 1
Since = = , we have 2 15! a b a ∑ 1 q = ab 15! a,b : √ √ a b = 15! ∑ ab = 15! a | 30420 ∑ 1 = a a | 30420 ( ) ( ) ( ) ( ) 1 1 1 1 1 1 1 1 1 1 = 1 + + + + + 1 + + + 1 + 1 + 2 4 8 16 32 3 9 27 5 7 ( ) ( ) ( ) ( ) 63 40 6 8 = 32 27 5 7 = 4 .