Difference between revisions of "2006 AMC 10B Problems/Problem 16"
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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. | 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. | ||
− | 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>. | + | 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 13:09, 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 | |
1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 • 16 • 17 • 18 • 19 • 20 • 21 • 22 • 23 • 24 • 25 | ||
All AMC 10 Problems and Solutions |
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