2000 AMC 12 Problems/Problem 18

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 1

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 $266$ but not $265$, for both days to be a Tuesday, year $N$ must be a leap year.

Hence, year $N-1$ is not a leap year, and so since there are \[265 + 300 = 565\] days between the date in years $N,\text{ }N-1$, this leaves a remainder of $5$ upon division by $7$. Since we are subtracting days, we count 5 days before Tuesday, which gives us $\boxed{\mathbf{(A)} \ \text{Thursday}.}$

Solution 2

The $300-29\cdot 7=97^{\text{th}}$ day of year $N$ and the $200-15\cdot 7=95^{\text{th}}$ day of year $N+1$ are Tuesdays. If there were the same number of days in year $N$ and year $N-1,$ the $99^{\text{th}}$ day of year $N-1$ will be a Tuesday. But year $N$ is a leap year because $97-95 \cong 366 \pmod{7},$ so the $98^{\text{th}}$ day of year $N-1$ is a Tuesday. It follows that the $100^{\text{th}}$ day of year $N-1$ is $\boxed{\mathbf{(A)} \ \text{Thursday}.}$

~dolphin7

See also

2000 AMC 12 (ProblemsAnswer KeyResources)
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 (ProblemsAnswer KeyResources)
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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