Difference between revisions of "2006 AMC 10B Problems/Problem 16"

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== Problem ==
 
== Problem ==
Leap Day, February 29, 2004, occured on a Sunday. On what day of the week will Leap Day, February 29, 2020, occur?  
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Leap Day, February <math>29</math>, <math>2000</math>, occurred on a Sunday. On what day of the week will Leap Day, February <math>29</math>, <math>2020</math>, occur?  
  
<math> \mathrm{(A) \ } \textrm{Tuesday} \qquad \mathrm{(B) \ } \textrm{Wednesday} \qquad \mathrm{(C) \ } \textrm{Thursday} \qquad \mathrm{(D) \ } \textrm{Friday} \qquad \mathrm{(E) \ } \textrm{Saturday} </math>
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<math> \textbf{(A) } \textrm{Tuesday} \qquad \textbf{(B) } \textrm{Wednesday} \qquad \textbf{(C) } \textrm{Thursday} \qquad \textbf{(D) } \textrm{Friday} \qquad \textbf{(E) } \textrm{Saturday} </math>
  
 
== Solution ==
 
== Solution ==
There are <math>365</math> days in a year, plus <math>1</math> 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).  
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There are <math>365</math> days in a year, plus <math>1</math> extra day if there is a Leap Day, which occurs on years that are multiples of <math>4</math> (with a few exceptions that don't affect this problem).  
  
Therefore, the number of days between Leap Day 2004 and Leap Day 2020 is:  
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Therefore, the number of days between Leap Day <math>2004</math> and Leap Day <math>2020</math> is:  
  
 
<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>.  
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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 <math>2020</math> occurs is the day of the week the day before Leap Day <math>2004</math> occurs, which is <math>\boxed{\textbf{(E) }\textrm{Saturday}}</math>.
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== Solution 2 (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> non-leap 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 \bmod{7} \equiv 6</math>, and six days after Sunday is <math>\boxed{\textbf{(E) }\textrm{Saturday}}</math>.
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*Note how we did not require any complex or time-consuming computation, and thus it will save crucial time, especially during a test like the AMC 10.
  
 
== See Also ==
 
== See Also ==
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[[Category:Introductory Number Theory Problems]]
 
[[Category:Introductory Number Theory Problems]]
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{{MAA Notice}}

Latest revision as of 02:41, 1 January 2024

Problem

Leap Day, February $29$, $2000$, occurred on a Sunday. On what day of the week will Leap Day, February $29$, $2020$, occur?

$\textbf{(A) } \textrm{Tuesday} \qquad \textbf{(B) } \textrm{Wednesday} \qquad \textbf{(C) } \textrm{Thursday} \qquad \textbf{(D) } \textrm{Friday} \qquad \textbf{(E) } \textrm{Saturday}$

Solution

There are $365$ days in a year, plus $1$ 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:

$16 \cdot 365 + 4 \cdot 1 = 5844$

Since the days of the week repeat every $7$ days and $5844 \equiv -1 \bmod{7}$, the day of the week Leap Day $2020$ occurs is the day of the week the day before Leap Day $2004$ occurs, which is $\boxed{\textbf{(E) }\textrm{Saturday}}$.

Solution 2 (Feasible Shortcuts)

Since every non-leap year there is one day extra after the $52$ weeks, we can deduce that if we were to travel forward $x$ amount of non-leap years, then the answer would be "Sunday + $x$" 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 + $2x$" days.

Mixing these two together, we get $4$ leap years $(2008, 2012, 2016, 2020)$, and $12$ non-leap years. We thus get $12 + 2\cdot 4 = 20$ "extra days" after Sunday. Since seven days are in a week, we can get $20 \bmod{7} \equiv 6$, and six days after Sunday is $\boxed{\textbf{(E) }\textrm{Saturday}}$.

  • Note how we did not require any complex or time-consuming computation, and thus it will save crucial time, especially during a test like the AMC 10.

See Also

2006 AMC 10B (ProblemsAnswer KeyResources)
Preceded by
Problem 15
Followed by
Problem 17
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All AMC 10 Problems and Solutions

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