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

Answer

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

### Step-by-Step Solution:

1. **Identify the Count of Positive and Negative Integers**:
- We have 5 positive integers.
- We have 7 negative integers.

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

3. **Pairing the Integers**:
- We can pair the negative integers with positive integers. Since we have 5 positive integers, we can pair them with 5 of the 7 negative integers.
- This results in 5 pairs of one positive and one negative integer.

4. **Calculate the Result of the Pairs**:
- Each of these 5 pairs (one positive and one negative) will yield a negative product. Thus, the product of these 5 pairs is negative.

5. **Consider the Remaining Negative Integers**:
- After pairing, we have 2 negative integers left (since 7 - 5 = 2).
- The product of these 2 negative integers will be positive (as the product of two negative integers is positive).

6. **Combine the Results**:
- Now we have:
- A negative product from the 5 pairs (which is negative).
- A positive product from the remaining 2 negative integers (which is positive).
- When we multiply a negative product by a positive product, the result is negative.

### Final Answer:
The sign of the product when multiplying five positive integers and seven negative integers will be **negative**.

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