Difference between revisions of "2006 AMC 10B Problems/Problem 16"
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<math> 16 \cdot 365 + 4 \cdot 1 = 5844 </math> | <math> 16 \cdot 365 + 4 \cdot 1 = 5844 </math> | ||
− | Since the days of the week repeat every <math>7</math> days and <math> 5844 \equiv -1 \bmod{7}</math>, the day of the week Leap Day 2020 occurs is the day of the week the day before Leap Day 2004 occurs, which is <math>\textrm{Saturday} \Rightarrow E </math>. | + | Since the days of the week repeat every <math>7</math> days and <math> 5844 \equiv -1 \bmod{7}</math>, the day of the week Leap Day 2020 occurs is the day of the week the day before Leap Day 2004 occurs, which is <math>\textrm{Saturday} \Rightarrow E </math>. |
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+ | == Solution (Feasible Shortcuts) == | ||
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+ | Since every non-leap year there is one day extra after the <math>52</math> weeks, we can deduce that if we were to travel forward <math>x</math> amount of non-leap years, then the answer would be "Sunday + <math>x</math>" days. | ||
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+ | However, we also have leap-years in our pool of years traveled forward, and for every leap-year that passes, we have two extra days after the 52 weeks. So we would travel "Sunday + <math>2x</math>" days. | ||
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+ | Mixing these two together, we get <math>4</math> leap years <math>(2008, 2012, 2016, 2020)</math>, and <math>12</math> "normal years". We thus get <math>12 + 2\cdot 4 = 20</math> "extra days" after Sunday. Since seven days are in a week, we can get <math>20 \mod 7 = 6</math>, and six days after Sunday is <math>\textrm{Saturday} \Rightarrow E </math>. | ||
== See Also == | == See Also == |
Revision as of 13:08, 29 October 2017
Problem
Leap Day, February 29, 2004, occurred on a Sunday. On what day of the week will Leap Day, February 29, 2020, occur?
Solution
There are days in a year, plus extra day if there is a Leap Day, which occurs on years that are multiples of 4 (with a few exceptions that don't affect this problem).
Therefore, the number of days between Leap Day 2004 and Leap Day 2020 is:
Since the days of the week repeat every days and , the day of the week Leap Day 2020 occurs is the day of the week the day before Leap Day 2004 occurs, which is .
Solution (Feasible Shortcuts)
Since every non-leap year there is one day extra after the weeks, we can deduce that if we were to travel forward amount of non-leap years, then the answer would be "Sunday + " days.
However, we also have leap-years in our pool of years traveled forward, and for every leap-year that passes, we have two extra days after the 52 weeks. So we would travel "Sunday + " days.
Mixing these two together, we get leap years , and "normal years". We thus get "extra days" after Sunday. Since seven days are in a week, we can get , and six days after Sunday is .
See Also
2006 AMC 10B (Problems • Answer Key • Resources) | ||
Preceded by Problem 15 |
Followed by Problem 17 | |
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All AMC 10 Problems and Solutions |
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