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

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Leap Day, February 29, 2004, occured on a Sunday. On what day of the week will Leap Day, February 29, 2020, occur?  
 
Leap Day, February 29, 2004, occured on a Sunday. On what day of the week will Leap Day, February 29, 2020, occur?  
  
<math> \mathrm{(A) \ } Tuesday \qquad \mathrm{(B) \ } Wednesday \qquad \mathrm{(C) \ } Thursday \qquad \mathrm{(D) \ } Friday \qquad \mathrm{(E) \ } Saturday </math>
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<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>
  
 
== Solution ==
 
== Solution ==
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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>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 2020 occurs is the day of the week the day before Leap Day 2004 occurs, which is <math>\textrm{Saturday} \Rightarrow E </math>.  
  
 
== See Also ==
 
== See Also ==
 
*[[2006 AMC 10B Problems]]
 
*[[2006 AMC 10B Problems]]
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*[[2006 AMC 10B Problems/Problem 15|Previous Problem]]
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*[[2006 AMC 10B Problems/Problem 17|Next Problem]]

Revision as of 15:02, 2 August 2006

Problem

Leap Day, February 29, 2004, occured on a Sunday. On what day of the week will Leap Day, February 29, 2020, occur?

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

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 $\textrm{Saturday} \Rightarrow E$.

See Also

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