HMMT 二月 2001 · ADV 赛 · 第 5 题
HMMT February 2001 — ADV Round — Problem 5
题目详情
英文原题
- Find the 6-digit number beginning and ending in the digit 2 that is the product ofthree consecutive even integers.
解析
英文解析
- Find the 6-digit number beginning and ending in the digit 2 that is the product ofthree consecutive even integers.
Solution: Because the last digit of the product is 2, none of the three consecutive evenintegers end in 0. Thus they must end in 2, 4, 6 or 4, 6, 8, so they must end in 4, 6, 8 since 2 · 4 · 6
does not end in 2. Call the middle integer n . Then the product is ( n − 2) n ( n + 2) = n − 4 n ,3
√ √
√ √
3 3
3 3
3 3
so n > 200000 = 200 · 10 ≈ 60, but clearly n < 300000 = 300 · 10 < 70. Thusn = 66, and the product is 66 − 4 · 66 = 287232 .3