AMC 8 2017 Problem 23
AMC8 2017 年第 23 题
整除与因数★★★★☆
Each day for four days, Linda traveled for one hour at a speed that resulted in her traveling one mile in an integer number of minutes. Each day after the first, her speed decreased so that the number of minutes to travel one mile increased by 5 minutes over the preceding day. Each of the four days, her distance traveled was also an integer number of miles. What was the total number of miles for the four trips?
四天里的每一天，Linda 每天都旅行 1 小时，并且每天的旅行速度都满足这样的要求：保证旅行 1 英里所花的分钟数都是整数。第一天之后的每一天，她的速度不断降低，使得旅行 1 英里后一天所花的分钟数比前一天多 5 分钟。这 4 天的每一天，她所旅行的路程也都是个整数。那么这 4 天的总路程是多少英里？
- A.10
- B.15
- C.25
- D.50
- E.82
正确答案 · Correct Answer
C
解析 · Solution
解法1
常规解法 · 每英里分钟须整除 60
里程 =60/t 为整数 ⇒ 「每英里分钟」 整除 60。四天 t,t+5,t+10,t+15 都整除 60。60 的因数中公差 5 的四连只有 5,10,15,20。里程 12+6+4+3=25。
巧解 · 锁等差因数链
必整除 60 且四天成公差 5 的等差 ⇒ 唯一链 5,10,15,20。总里程 。
Solution 1
Elementary · minutes per mile must divide 60
Each day's distance is 60/t, which must be an integer, so the minutes per mile divides 60. Over four days t,t+5,t+10,t+15 are all divisors of 60; the only four divisors of 60 in arithmetic progression with common difference 5 are 5,10,15,20. The total distance is miles.
Slick · lock the AP divisor chain
must divide 60 while the four days form an arithmetic progression with common difference 5, which forces the unique chain 5,10,15,20. The total is .
继续练习