8) Ivan packed 56 apples and 72 pears into boxes. He packed both apples and pears into each box. He put the same number of apples into every box and the same number of pears into every box. He sold each box for $$ \$ 19.95 $$. What is the greatest amount of money Ivan could earn selling all the boxes of fruit? (Hint: Two part problem. First, find out how mar boxes Ivan can make.)

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To determine the greatest amount of money Ivan could earn from selling all the boxes of fruit, we'll break the problem down into two parts:

1. **Determine the Maximum Number of Boxes Ivan Can Make**
2. **Calculate the Total Earnings Based on the Number of Boxes**

### **1. Determine the Maximum Number of Boxes Ivan Can Make**

Ivan has:
- **56 apples**
- **72 pears**

He wants to pack both apples and pears into each box with:
- The **same number of apples** in every box
- The **same number of pears** in every box

To maximize the number of boxes, we need to find the **greatest common divisor (GCD)** of 56 and 72. The GCD will give us the largest number of boxes that can be made where each box has the same number of apples and pears.

**Step-by-Step Calculation:**

- **Find the GCD of 56 and 72:**
  - **Prime factorization of 56:** $$ 56 = 2 \times 2 \times 2 \times 7 $$
  - **Prime factorization of 72:** $$ 72 = 2 \times 2 \times 2 \times 3 \times 3 $$
  
- **Identify the common factors:**
  - Both numbers have three 2's in their prime factorization.
  
- **Calculate the GCD:**
  - $$ \text{GCD} = 2 \times 2 \times 2 = 8 $$
  
So, **Ivan can make a maximum of 8 boxes**.

### **2. Calculate the Total Earnings**

Ivan sells each box for **\$19.95**. To find the total earnings:

**Total Earnings = Number of Boxes × Price per Box**

- Number of Boxes = 8
- Price per Box = \$19.95

**Calculation:**
$$
8 \times \$19.95 = \$159.60
$$

### **Conclusion**

The greatest amount of money Ivan could earn by selling all the boxes of fruit is **\$159.60**.
Ivan can make 8 boxes and earn a total of \$159.60 by selling all the boxes of fruit.

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