HMMT 二月 2018 · 冲刺赛 · 第 10 题
HMMT February 2018 — Guts Round — Problem 10
题目详情
- [ 7 ] Let a positive integer n be called a cubic square if there exist positive integers a, b with n = 2 3 gcd( a , b ). Count the number of cubic squares between 1 and 100 inclusive.
解析
- [ 7 ] Let a positive integer n be called a cubic square if there exist positive integers a, b with n = 2 3 gcd( a , b ). Count the number of cubic squares between 1 and 100 inclusive. Proposed by: Ashwin Sah Answer: 13 This is easily equivalent to v ( n ) 6 ≡ 1 , 5 (mod 6) for all primes p . We just count: p ≥ 11 = ⇒ v ( n ) = 1 p p is clear, so we only look at the prime factorizations with primes from { 2 , 3 , 5 , 7 } . This is easy to compute: we obtain 13.