HMMT 二月 2003 · 冲刺赛 · 第 8 题
HMMT February 2003 — Guts Round — Problem 8
题目详情
- [6] How many solutions in nonnegative integers ( a, b, c ) are there to the equation a b 2 + 2 = c ! ?
解析
- How many solutions in nonnegative integers ( a, b, c ) are there to the equation a b 2 + 2 = c ! ? Solution: 5 a b We can check that 2 + 2 is never divisible by 7, so we must have c < 7. The binary a b representation of 2 + 2 has at most two 1’s. Writing 0! , 1! , 2! , . . . , 6! in binary, we can check that the only possibilities are c = 2 , 3 , 4, giving solutions (0 , 0 , 2) , (1 , 2 , 3) , (2 , 1 , 3) , (3 , 4 , 4) , (4 , 3 , 4).