2013 AMC 10A Problems/Problem 17
Daphne is visited periodically by her three best friends: Alice, Beatrix, and Claire. Alice visits every third day, Beatrix visits every fourth day, and Claire visits every fifth day. All three friends visited Daphne yesterday. How many days of the next -day period will exactly two friends visit her?
The -day time period can be split up into , -day time periods, because after days, all three of them visit again (Least common multiple of , , and ). You can find how many times each pair of visitors can meet by finding the LCM of their visiting days and dividing that number by 60. Remember to subtract , because you do not wish to count the th day, when all three visit.
A and B visit times.
A and C visit times.
B and C visit times.
This is a total of visits per day period. Therefore, the total number of -person visits is .
- Note: We do not have to worry about the numbers over 360: () having 2 factors. This is because we can rewrite
. We note that is not further divisible by 4 or 5, is not further divisible by 3 or 5, 73 is not further divisible by 3 or 4. Therefore, none of the numbers from have 2 factors of or , so we can conclude that the answer is indeed
From the information above, we get that
Now, we want the days in which exactly two of these people meet up
The three pairs are , , .
Notice that we are trying to find the LCM of each pair.
Hence, , ,
Notice that we want to eliminate when all these friends meet up. By doing this, we will find the LCM of the three letters.
Now, we add all of the days up(including overcount).
We get . Now, because , we have to subtract days from every pair. Hence, our answer is .
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