Question: 48
15\times20
=300
A baker is packing some bread for delivery. He can either pack them into a box of 15 or 20 bread loaves. What is the minimum number of bread loaves he has for packing?
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 again?
1. **Problem 1: Minimum number of bread loaves for packing**
The baker can pack bread in boxes of $15$ or $20$ loaves. We want to find the minimum number of bread loaves that can be packed exactly into both box sizes.
2. **Formula and concept:**
This is a problem of finding the Least Common Multiple (LCM) of $15$ and $20$ because the minimum number divisible by both $15$ and $20$ is their LCM.
3. **Calculate the LCM:**
- Prime factorization:
- $15 = 3 \times 5$
- $20 = 2^2 \times 5$
- LCM takes the highest powers of all primes:
$$\text{LCM}(15,20) = 2^2 \times 3 \times 5 = 4 \times 3 \times 5 = 60$$
4. **Answer for problem 1:**
The minimum number of bread loaves is $60$.
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5. **Problem 2: Tom and Jerry cleaning schedule**
Tom cleans every $12$ days, Jerry every $16$ days. They start cleaning today (day $0$). We want to find the next day they both clean together.
6. **Formula and concept:**
Again, this is an LCM problem. The day they both clean together again is the LCM of $12$ and $16$.
7. **Calculate the LCM:**
- Prime factorization:
- $12 = 2^2 \times 3$
- $16 = 2^4$
- LCM takes highest powers:
$$\text{LCM}(12,16) = 2^4 \times 3 = 16 \times 3 = 48$$
8. **Answer for problem 2:**
They will both clean their rooms again on day $48$.
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**Final answers:**
- Minimum bread loaves: $60$
- Both clean again on day: $48$