Skip to main content

2012 AMC 10A Problem 12

Problem 12 of 25IntermediateAlgebraNumber 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 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?

Answer choices

Show solution

Solution

On a typical year with 365365 days, moving one year into the future moves the day of the week forward one since 365=527+1. 365 = 52 \cdot 7 + 1. On a leap year however, the day of the week gets moved forward twice since there is an extra day. Fifty of the years between 20122012 and 18121812 are multiples of 4,4, but we have to discard 1900,1900, which leaves us with 4949 leap years. Therefore, moving back 200200 years means we have to go back 200+49=249=357+4 200 + 49 = 249 = 35 \cdot 7 + 4 days, which means we move back 44 days in the week. This takes us back to Friday, which is the day that corresponds to February 7,7, 1812.1812. Thus, A is the correct answer.

More practice

Concepts: date and time · modular arithmetic

Problem text and solution from the LIVE past-contest archive. See also the AoPS wiki page for community solutions.