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Let's break down and solve the two questions from the image:

### Question 31:
In a shop, there are 36 oranges, 72 apples, and 48 pears. The shop manager wants to pack baskets such that each basket has the same number of oranges, apples, and pears. At most, how many baskets can be packed?

To find out how many baskets can be packed, we need to find the **Greatest Common Divisor (GCD)** of 36, 72, and 48.

- The factors of 36 are: 1, 2, 3, 4, 6, 9, 12, 18, 36.
- The factors of 72 are: 1, 2, 3, 4, 6, 8, 9, 12, 18, 24, 36, 72.
- The factors of 48 are: 1, 2, 3, 4, 6, 8, 12, 16, 24, 48.

The largest number common to all these is **12**.

Thus, **12 baskets** can be packed, with each basket containing:
- \( \frac{36}{12} = 3 \) oranges,
- \( \frac{72}{12} = 6 \) apples, and
- \( \frac{48}{12} = 4 \) pears.

### Question 32:
Buses of three bus routes arrive at a bus stop every 25, 30, and 45 minutes respectively. If buses of these three bus routes have arrived at the bus stop at 9 a.m., what will be the next time that they arrive at the bus stop at the same time again?

We need to find the **Least Common Multiple (LCM)** of 25, 30, and 45 to determine when the buses will next arrive together.

- Prime factorization of 25: \( 5^2 \),
- Prime factorization of 30: \( 2 \times 3 \times 5 \),
- Prime factorization of 45: \( 3^2 \times 5 \).

To find the LCM, we take the highest power of each prime:
- \( 2^1 \) (from 30),
- \( 3^2 \) (from 45),
- \( 5^2 \) (from 25).

Thus, the LCM is:
\[
LCM = 2^1 \times 3^2 \times 5^2 = 2 \times 9 \times 25 = 450 \, \text{minutes}.
\]

450 minutes equals **7 hours and 30 minutes**. So, the buses will next arrive together at:
\[
9:00 \, \text{a.m.} + 7 \, \text{hours and} \, 30 \, \text{minutes} = 4:30 \, \text{p.m.}
\]

The next time the buses arrive together is at **4:30 p.m.**.

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Would you like further details on either solution?

Here are 5 related questions:
1. What is the GCD of three other numbers like 24, 36, and 60?
2. How is the LCM useful in real-world problems like planning schedules?
3. How do you calculate the GCD of more than two numbers?
4. How would the solution change in Question 32 if the buses arrived every 20, 35, and 50 minutes?
5. Can the GCD method be applied to solve problems involving ratios?

**Tip:** When dealing with multiple quantities and uniform groupings, GCD helps with dividing items into equal parts, while LCM helps in synchronizing events that happen repeatedly.

31. In a shop, there are 36 oranges, 72 apples, and 48 pears. The shop manager wants to pack baskets such that each basket has the same numbers of oranges, apples, and pears. At most how many baskets can be packed?
32. Buses of three bus routes arrive at a bus stop every 25, 30, and 45 minutes respectively. If buses of these three bus routes have arrived at the bus stop at 9 a.m., what will be the next time that they arrive at the bus stop at the same time again?

Learn how to solve a problem involving the GCD of 36, 72, and 48 to pack baskets with equal numbers of fruits, and how to find the next time buses arrive together using the LCM of 25, 30, and 45. Suitable for students in Grades 7-9.

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