HMMT 二月 2007 · GEN1 赛 · 第 5 题
HMMT February 2007 — GEN1 Round — Problem 5
题目详情
- [ 4 ] Find the smallest positive integer that is twice a perfect square and three times a perfect cube. n n
解析
- [ 4 ] Find the smallest positive integer that is twice a perfect square and three times a perfect cube. Answer: 648 . Let n be such a number. If n is divisible by 2 and 3 exactly e and e times, then 2 3 e is odd and a multiple of three, and e is even and one more than a multiple of three. The smallest 2 3 3 4 possible exponents are n = 3 and n = 4. The answer is then 2 · 3 = 648 . 2 3 n n