AMC8 2019
AMC8 2019 · Q14
AMC8 2019 · Q14. It mainly tests Logic puzzles, Remainders & modular arithmetic.
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?
伊莎贝拉有$6$张可在Pete's Sweet Treats兑换免费冰淇淋甜筒的优惠券。为了让优惠券用得久一些，她决定每$10$天兑换一张，直到用完。她知道Pete's在周日休息，但当她在日历上圈出$6$个日期时，她意识到没有一个圈出的日期落在周日。那么伊莎贝拉第一张优惠券是在星期几兑换的？
(A)
\text{Monday}
\text{Monday}
(B)
\text{Tuesday}
\text{Tuesday}
(C)
\text{Wednesday}
\text{Wednesday}
(D)
\text{Thursday}
\text{Thursday}
(E)
\text{Friday}
\text{Friday}
Answer
Correct choice: (C)
正确答案：（C）
Solution
Let $\text{Day }1$ to $\text{Day\\ }2$ denote a day where one coupon is redeemed and the day when the second coupon is redeemed.
If she starts on a $\text{Monday}$, she redeems her next coupon on $\text{Thursday}$.
$\text{Thursday}$ to $\text{Sunday}$.
Thus, $\textbf{(A)}\ \text{Monday}$ is incorrect.
If she starts on a $\text{Tuesday}$, she redeems her next coupon on $\text{Friday}$.
$\text{Friday}$ to $\text{Monday}$.
$\text{Monday}$ to $\text{Thursday}$.
$\text{Thursday}$ to $\text{Sunday}$.
Thus $\textbf{(B)}\ \text{Tuesday}$ is incorrect.
If she starts on a $\text{Wednesday}$, she redeems her next coupon on $\text{Saturday}$.
$\text{Saturday}$ to $\text{Tuesday}$.
$\text{Tuesday}$ to $\text{Friday}$.
$\text{Friday}$ to $\text{Monday}$.
$\text{Monday}$ to $\text{Thursday}$.
And on $\text{Thursday}$, she redeems her last coupon.
No Sunday occured; thus, $\boxed{\textbf{(C)}\ \text{Wednesday}}$ is correct.
Checking for the other options,
If she starts on a $\text{Thursday}$, she redeems her next coupon on $\text{Sunday}$.
Thus, $\textbf{(D)}\ \text{Thursday}$ is incorrect.
If she starts on a $\text{Friday}$, she redeems her next coupon on $\text{Monday}$.
$\text{Monday}$ to $\text{Thursday}$.
$\text{Thursday}$ to $\text{Sunday}$.
Checking for the other options gave us negative results; thus, the answer is $\boxed{\textbf{(C)}\ \text{Wednesday}}$.
设Day 1为第一张兑换日，Day 2为第二张兑换日。
如果从周一开始，下一个是周四。
周四到周日。
因此，周一错误。
如果从周二开始，下一个是周五。
周五到周一。
周一到周四。
周四到周日。
因此周二错误。
如果从周三开始，下一个是周六。
周六到周二。
周二到周五。
周五到周一。
周一到周四。
周四兑换最后一张。
没有周日，因此$\boxed{\textbf{(C)}\ \text{Wednesday}}$正确。
其他选项：
从周四开始，下一个是周日，因此错误。
从周五开始，下一个是周一，周一到周四，周四到周日，因此错误。
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