What will be the sign of the product if we multiply:
six positive and eight negative integers?

Answer

To determine the sign of the product when multiplying six positive integers and eight negative integers, we can follow these steps:

### Step-by-Step Solution:

1. **Identify the Count of Positive and Negative Integers**:
- We have 6 positive integers.
- We have 8 negative integers.

2. **Understand the Rules of Multiplication**:
- The product of two positive integers is positive.
- The product of two negative integers is positive.
- The product of a positive integer and a negative integer is negative.

3. **Pair the Negative Integers**:
- Since we have 8 negative integers, we can pair them up:
- Pair 1: Negative 1 and Negative 2 (result is positive)
- Pair 2: Negative 3 and Negative 4 (result is positive)
- Pair 3: Negative 5 and Negative 6 (result is positive)
- Pair 4: Negative 7 and Negative 8 (result is positive)
- This results in 4 pairs of negative integers, and since each pair results in a positive product, we have 4 positive products from the negative integers.

4. **Combine Positive Products**:
- Now, we have 4 positive products from the negative pairs.
- We also have 6 positive integers.
- When we multiply these positive products with the positive integers, the result remains positive.

5. **Final Calculation of Signs**:
- Since we have an even number of negative integers (8), all of them can be paired, resulting in a positive product.
- The multiplication of positive integers (from the 6 positive integers) with the positive product from the negative pairs will yield a positive result.

### Conclusion:
The sign of the product when multiplying six positive integers and eight negative integers is **positive**.

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