Question: The museum performs shows in schools every $Monday$ and in public libraries every fifth day (in both weekdays and weekends). If the museum did both a school show and a library show on Monday, how many days will it be until it does both shows on the same day again?
1. **State the problem:**
We want to find the number of days until the museum performs both a school show (which happens every Monday) and a library show (which happens every 5 days) on the same day again, given that they both happened on the same Monday.
2. **Identify the cycles:**
- School shows occur every 7 days (since Monday repeats weekly).
- Library shows occur every 5 days.
3. **Find the least common multiple (LCM):**
The problem reduces to finding the LCM of the two cycles, $7$ and $5$, because the shows coincide when both cycles align.
4. **Calculate the LCM:**
- Prime factors of $7$ are $7$.
- Prime factors of $5$ are $5$.
- Since $7$ and $5$ are coprime, $$\text{LCM}(7,5) = 7 \times 5 = 35.$$
5. **Interpretation:**
The museum will perform both shows on the same day again after $35$ days.
**Final answer:**
$$35$$ days.