# Problem of the Day

Aaron, Ben, and Cliff all make resolutions to exercise more in the new year. Aaron resolves to exercise on every fourth day, Ben resolves to exercise on every seventh day, and Cliff resolves to exercise on every tenth day. If they follow their resolutions exactly, what is the probability that they all exercised on a randomly selected day, assuming this is not a leap year?

A.

B.

C.

D.

E.

### Solution

[latexpage]

To determine how often they exercise on the same day, first calculate the least common multiple of their individual exercise frequencies: LCM(4,7,10)=140. This means they all exercise on the same day every 140th day. So they would exercise together on the 140th and 280th day of the year. Thus there is a $\frac{2}{365}$ chance of them all exercising on a randomly selected day of the year.

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