Question 2

Explanation

To determine when all three suppliers deliver vegetables simultaneously, we use the concept of the Least Common Multiple (LCM) of their delivery schedules, which are $x$, $4$, and $7$.

It is given that the $16$-th simultaneous delivery occurs $1260$ days after the first delivery. This means that $15$ simultaneous delivery intervals have passed since the first meeting.

We can calculate the LCM of the three numbers by dividing the total number of days by the number of intervals occurred:

So, the LCM of $x$, $4$, and $7$ is $84$. We know that the LCM of $4$ and $7$ is $28$. Therefore, we must find the value of $x$ such that the LCM of $x$ and $28$ is $84$.

Let's check the available options:

• If $x = 6$, then LCM($6, 28$) = $84$. (Correct)
• If $x = 8$, then LCM($8, 28$) = $56$. (Incorrect)
• If $x = 9$, then LCM($9, 28$) = $252$. (Incorrect)
• If $x = 10$, then LCM($10, 28$) = $140$. (Incorrect)
• If $x = 11$, then LCM($11, 28$) = $308$. (Incorrect)

Thus, the value of $x$ that satisfies the condition is $6$.