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Difference between revisions of "2000 AMC 12 Problems/Problem 18"

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{{duplicate|[[2000 AMC 12 Problems|2000 AMC 12 #18]] and [[2000 AMC 10 Problems|2000 AMC 10 #25]]}}
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== Problem ==
 
== Problem ==
 
In year <math>N</math>, the <math>300^{\text{th}}</math> day of the year is a Tuesday. In year <math>N+1</math>, the <math>200^{\text{th}}</math> day is also a Tuesday. On what day of the week did the <math>100</math><sup>th</sup> day of year <math>N-1</math> occur?
 
In year <math>N</math>, the <math>300^{\text{th}}</math> day of the year is a Tuesday. In year <math>N+1</math>, the <math>200^{\text{th}}</math> day is also a Tuesday. On what day of the week did the <math>100</math><sup>th</sup> day of year <math>N-1</math> occur?
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== See also ==
 
== See also ==
 
{{AMC12 box|year=2000|num-b=17|num-a=19}}
 
{{AMC12 box|year=2000|num-b=17|num-a=19}}
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{{AMC10 box|year=2000|num-b=24|after=Last Problem}}
  
 
[[Category:Introductory Number Theory Problems]]
 
[[Category:Introductory Number Theory Problems]]

Revision as of 23:12, 26 November 2011

The following problem is from both the 2000 AMC 12 #18 and 2000 AMC 10 #25, so both problems redirect to this page.

Problem

In year $N$, the $300^{\text{th}}$ day of the year is a Tuesday. In year $N+1$, the $200^{\text{th}}$ day is also a Tuesday. On what day of the week did the $100$th day of year $N-1$ occur?

$\text {(A)}\ \text{Thursday} \qquad \text {(B)}\ \text{Friday}\qquad \text {(C)}\ \text{Saturday}\qquad \text {(D)}\ \text{Sunday}\qquad \text {(E)}\ \text{Monday}$

Solution

There are either $65 + 200 = 265$ or $66 + 200 = 266$ days between the first two dates depending upon whether or not year $N$ is a leap year. Since $7$ divides into $266$, then it is possible for both dates to be Tuesday; hence year $N$ is a leap year and $N-1$ is not a leap year. There are $265 + 300 = 565$ days between the date in years $N,N-1$, which leaves a remainder of $5$ upon division by $7$. Since we are subtracting days, we count 5 days before Tuesday, which gives us Thursday $\mathrm{(A)}$.

See also

2000 AMC 12 (Problems • Answer Key • Resources)
Preceded by
Problem 17 		Followed by
Problem 19
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 12 Problems and Solutions
2000 AMC 10 (Problems • Answer Key • Resources)
Preceded by
Problem 24 		Followed by
Last Problem
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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