AMC 8 2019 Problem 14
AMC8 2019 年第 14 题
余数与同余★★★☆☆
Isabella has 6 coupons that can be redeemed for free ice cream cones at Pete's Sweet Treats. In order to make the coupons last, she decides that she will redeem one every 10 days until she has used them all. She knows that Pete's is closed on Sundays, but as she circles the 6 dates on her calendar, she realizes that no circled date falls on a Sunday. On what day of the week does Isabella redeem her first coupon?
Isabella 有 6 张优惠券，可用于在 Pete 甜品店里免费兑换冰激凌。为了最后使用优惠券，她决定每 10 天兑换一次直到所有的优惠券都使用完毕。已知 Pete 甜品店周日是不营业的，当她在日历上圈出兑换冰激凌的这 6 天时，发现没有哪天是周日。问 Isabella 兑换她的第一个优惠券是在周几？
- A. / 周一
- B. / 周二
- C. / 周三
- D. / 周四
- E. / 周五
正确答案 · Correct Answer
C
解析 · Solution
Solution 1
Let the days of the week be numbers . 3 and 7 are coprime, so continuously adding 3 to a number will cycle through all numbers from 0 to 6. If a string of 6 numbers in this cycle does not contain 0, then if you minus 3 from the first number of this cycle, it will always be 0. So, the answer is. ~
Start counting on Sunday, to maximize the number of 10-day jumps before reaching Sunday again. Add the 10 ( ) days to reach the first coupon day on. ~
Let Sunday be Day 0, Monday be Day 1, Tuesday be Day 2, and so forth. We see that Sundays fall on Day , where n is a multiple of seven. If Isabella starts using her coupons on Monday (Day 1), she will fall on a Day that is a multiple of seven, a Sunday (her third coupon will be "used" on Day 21). Similarly, if she starts using her coupons on Tuesday (Day 2), Isabella will fall on a Day that is a multiple of seven (Day 42). Repeating this process, if she starts on Wednesday (Day 3), Isabella will first fall on a Day that is a multiple of seven, Day 63 (13, 23, 33, 43, 53 are not multiples of seven), but on her seventh coupon, of which she only has six. So, the answer is .
There is a ten day gap, so every time you are 3 days ahead. For Sunday to be the 7th day you must be 6 days ahead, so 6x3 = 18. 18 days prior is the same as 4 days prior to Sunday which is .
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