AMC10 2012 A

AMC10 2012 A · Q12

AMC10 2012 A · Q12. It mainly tests Divisibility & factors, Remainders & modular arithmetic.

A year is a leap year if and only if the year number is divisible by 400 (such as 2000) or is divisible by 4 but not by 100 (such as 2012). The 200th anniversary of the birth of novelist Charles Dickens was celebrated on February 7, 2012, a Tuesday. On what day of the week was Dickens born?

一年是闰年当且仅当年份能被 400 整除（如 2000 年）或能被 4 整除但不能被 100 整除（如 2012 年）。小说家查尔斯·狄更斯的诞生 200 周年纪念于 2012 年 2 月 7 日星期二庆祝。狄更斯出生于星期几？

(A) Friday 星期五

(B) Saturday 星期六

(D) Monday 星期一

(E) Tuesday 星期二

Answer

Correct choice: (A)

正确答案：（A）

Solution

There were $200\cdot365 = 73000$ non-leap days in the 200-year time period from February 7, 1812 to February 7, 2012. One fourth of those years contained a leap day, except for 1900, so there were $1/4 \cdot 200 - 1 = 49$ leap days during that time. Therefore Dickens was born 73049 days before a Tuesday. Because the same day of the week occurs every 7 days and $73049 = 7\cdot10435 + 4$, the day of Dickens’ birth (February 7, 1812) was 4 days before a Tuesday, which was a Friday.

从 1812 年 2 月 7 日到 2012 年 2 月 7 日的 200 年期间，有 $200\cdot365 = 73000$ 个非闰日。其中四分之一的年份含有闰日，但 1900 年除外，因此共有 $1/4 \cdot 200 - 1 = 49$ 个闰日。所以狄更斯出生于星期二前 73049 天。因为一周 7 天重复，且 $73049 = 7\cdot10435 + 4$，故 1812 年 2 月 7 日是星期二前 4 天，即星期五。

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