Difference between revisions of "2010 AMC 12B Problems/Problem 4"
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{{duplicate|[[2010 AMC 12B Problems|2010 AMC 12B #4]] and [[2010 AMC 10B Problems|2010 AMC 10B #5]]}} | {{duplicate|[[2010 AMC 12B Problems|2010 AMC 12B #4]] and [[2010 AMC 10B Problems|2010 AMC 10B #5]]}} | ||
| − | == Problem | + | == Problem == |
| − | A month with <math>31</math> days has the same number of Mondays and Wednesdays.How many of the seven days of the week could be the first day of this month? | + | A month with <math>31</math> days has the same number of Mondays and Wednesdays. How many of the seven days of the week could be the first day of this month? |
<math>\textbf{(A)}\ 2 \qquad \textbf{(B)}\ 3 \qquad \textbf{(C)}\ 4 \qquad \textbf{(D)}\ 5 \qquad \textbf{(E)}\ 6</math> | <math>\textbf{(A)}\ 2 \qquad \textbf{(B)}\ 3 \qquad \textbf{(C)}\ 4 \qquad \textbf{(D)}\ 5 \qquad \textbf{(E)}\ 6</math> | ||
| Line 8: | Line 8: | ||
== Solution == | == Solution == | ||
<math>31 \equiv 3 \pmod {7}</math> so the week cannot start with Saturday, Sunday, Tuesday or Wednesday as that would result in an unequal number of Mondays and Wednesdays. Therefore, Monday, Thursday, and Friday are valid so the answer is <math>\boxed{B}</math>. | <math>31 \equiv 3 \pmod {7}</math> so the week cannot start with Saturday, Sunday, Tuesday or Wednesday as that would result in an unequal number of Mondays and Wednesdays. Therefore, Monday, Thursday, and Friday are valid so the answer is <math>\boxed{B}</math>. | ||
| + | |||
| + | ==Video Solution== | ||
| + | https://youtu.be/4u7WqHb42M0 | ||
| + | |||
| + | ~Education, the Study of Everything | ||
| + | |||
| + | ==Video Solution== | ||
| + | https://youtu.be/uAc9VHtRRPg?t=329 | ||
| + | |||
| + | ~IceMatrix | ||
== See also == | == See also == | ||
{{AMC12 box|year=2010|num-b=3|num-a=5|ab=B}} | {{AMC12 box|year=2010|num-b=3|num-a=5|ab=B}} | ||
| + | {{AMC10 box|year=2010|num-b=4|num-a=6|ab=B}} | ||
{{MAA Notice}} | {{MAA Notice}} | ||
Latest revision as of 10:24, 3 August 2022
- The following problem is from both the 2010 AMC 12B #4 and 2010 AMC 10B #5, so both problems redirect to this page.
Problem
A month with 
days has the same number of Mondays and Wednesdays. How many of the seven days of the week could be the first day of this month?
Solution
so the week cannot start with Saturday, Sunday, Tuesday or Wednesday as that would result in an unequal number of Mondays and Wednesdays. Therefore, Monday, Thursday, and Friday are valid so the answer is 
.
Video Solution
~Education, the Study of Everything
Video Solution
https://youtu.be/uAc9VHtRRPg?t=329
~IceMatrix
See also
| 2010 AMC 12B (Problems • Answer Key • Resources) | |
| Preceded by Problem 3 |
Followed by Problem 5 |
| 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 | |
| 2010 AMC 10B (Problems • Answer Key • Resources) | ||
| Preceded by Problem 4 |
Followed by Problem 6 | |
| 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 | ||
The problems on this page are copyrighted by the Mathematical Association of America's American Mathematics Competitions. 
