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, 2021

Number theory Difficulty 2.6 Junior Find the answer Canada

Trains arrive at Pascal Station every $x$ minutes, where $x$ is a positive integer. Trains arrive at Pascal Station at many different times, including at 10:10 a.m., 10:55 a.m., and 11:58 a.m. Which of the following is a possible value of $x$ ?

Pick one

Solution

The length of time between 10:10 a.m. and 10:55 a.m. is 45 minutes.

The length of time between 10:55 a.m. and 11:58 a.m. is 1 hour and 3 minutes, or 63 minutes.

Since trains arrive at each of these times and we are told that trains arrive every $x$ minutes, then both 45 and 63 must be multiples of $x$ . (In other words, if we count forward repeatedly by $x$ minutes starting at 10:10 a.m., we will eventually count 10:55 a.m. and then eventually count 11:58 a.m.)

Of the given choices (9, 7, 5, 10, 11), only 9 is a factor of each of 45 and 63.

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Source: CEMC, University of Waterloo, licensed CC-BY-NC-4.0. Statement reproduced verbatim; metadata (topic, difficulty) added by this project.