Question: 16. Tom and Jerry are cleaning their rooms. Tom cleans his room every $12$ days, and Jerry cleans his room every $16$ days. If they both start cleaning their rooms today, on which day will they both clean their rooms on the same day again?
1. **State the problem:** Tom cleans his room every $12$ days and Jerry every $16$ days. They start cleaning today (day $0$). We want to find the next day when both clean their rooms on the same day.
2. **Formula and concept:** The problem asks for the least common multiple (LCM) of $12$ and $16$ because the LCM gives the smallest positive integer divisible by both numbers.
3. **Find prime factorizations:**
- $12 = 2^2 \times 3$
- $16 = 2^4$
4. **Calculate LCM:**
The LCM takes the highest powers of all primes:
$$\text{LCM}(12,16) = 2^4 \times 3 = 16 \times 3 = 48$$
5. **Interpretation:** They will both clean their rooms again on day $48$.
6. **Additional note:** If today is July 31, adding $48$ days leads to September 17 (assuming August has 31 days).
**Final answer:** They will both clean their rooms again on the same day after $\boxed{48}$ days.