PUMaC 2012 · 数论（B 组） · 第 3 题

PUMaC 2012 — Number Theory (Division B) — Problem 3

解析

[ 4 ] How many factors of 20 less than 20 are not factors of 20 ? ( ) 2 12 12 Solution: We start by writing the prime factorizations of 20 and 20 . 12 24 12 20 = 2 × 5 ( ) 2 12 48 24 20 = 2 × 5 ( ) 2 12 12 12 20 has 49 × 25 = 1225 factors, and 20 has 25 × 13 = 325 factors. Aside from 20 , each ( ) 2 12 12 12 factor of 20 that is greater than 20 can be paired up with a factor that is less than 20 , 1 ( ) 2 12 12 so 20 has 612 factors less than 20 . Then we see that 612 − 324 = 288 of those factors 12 are not factors of 20 .