2000 AMC 12 Problems/Problem 18

From AoPSWiki

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?

$\text {(A)}\ \text{Thursday} \qquad \text {(B)}\ \text{Friday}\qquad \text {(C)}\ \text{Saturday}\qquad \text {(D)}\ \text{Su...$

Solution

There are either $165 + 200 = 265$ or $166 + 200 = 266$ 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 $\mathrm{(A)}$ .

2000 AMC 12 (Problems • 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

Personal tools

Navigation

Search

Toolbox

2000 AMC 12 Problems/Problem 18

From AoPSWiki

Problem

Solution

See also