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2012 AMC 12A Problem 9

Problem 9 of 25EasierAlgebraNumber Theory

A year is a leap year if and only if the year number is divisible by 400400 (such as 20002000) or is divisible by 44 but not by 100100 (such as 20122012). The 200200th anniversary of the birth of novelist Charles Dickens was celebrated on February 7,7, 2012,2012, a Tuesday. On what day of the week was Dickens born?

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Solution

From February 7,7, 18121812 to February 7,7, 20122012 there are 200365=73000200 \cdot 365 = 73000 ordinary days plus one for each leap day. One quarter of the 200200 years contain a leap day, except 1900,1900, giving 142001=49\tfrac14 \cdot 200 - 1 = 49 leap days. So the span is 7304973049 days. Since 73049=710435+4,73049 = 7 \cdot 10435 + 4, the birth day was 44 days before a Tuesday, which is a Friday. Thus, the correct answer is A.

More practice

Concepts: modular arithmetic · date and time

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