Question 1

Explanation

Given the start time is $07:00$ and the observation ends at $09:24$. Let's calculate the total duration in minutes:

It is stated that the three runners passed the start line together $4$ times during this interval. This means they meet every:

This meeting interval corresponds to the Least Common Multiple (LCM) of the runners' lap times.

We need to find the value of $x$ such that $\text{LCM}(12, x, 9) = 36$. First, find the LCM of $12$ and $9$:

Since $\text{LCM}(12, 9)$ is already $36$, for $\text{LCM}(12, 9, x)$ to remain $36$, $x$ must be a factor of $36$ or a number that, when combined with $12$ and $9$, results in an LCM of $36$. Let's check the options:

• If $x = 5$: $\text{LCM}(36, 5) = 180$ (Incorrect)
• If $x = 6$: $\text{LCM}(36, 6) = 36$ (Correct)
• If $x = 7$: $\text{LCM}(36, 7) = 252$ (Incorrect)
• If $x = 10$: $\text{LCM}(36, 10) = 180$ (Incorrect)
• If $x = 13$: $\text{LCM}(36, 13) = 468$ (Incorrect)

Thus, the satisfying value for $x$ is $6$.