2010 AMC 12B Problems/Problem 4

The following problem is from both the 2010 AMC 12B #4 and 2010 AMC 10B #5, so both problems redirect to this page.

Problem 4

A month with $31$ 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?

$\textbf{(A)}\ 2 \qquad \textbf{(B)}\ 3 \qquad \textbf{(C)}\ 4 \qquad \textbf{(D)}\ 5 \qquad \textbf{(E)}\ 6$

Solution

$31 \equiv 3 \pmod {7}$ 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 $\boxed{B}$ .

Video Solution

https://youtu.be/uAc9VHtRRPg?t=329

~IceMatrix

2010 AMC 12B (Problems • Answer Key • Resources)
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All AMC 12 Problems and Solutions

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2010 AMC 12B Problems/Problem 4

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Problem 4

Solution

Video Solution

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