To answer these questions, we need to consider the properties of multiplying negative and positive integers. Specifically, the sign of the product depends on the number of negative integers involved in the multiplication.
### Part (a)
Question: What will be the sign if you multiply 105 negative integers and 3 positive integers?
Step-by-Step Solution:
1. Number of Negative Integers: There are 105 negative integers.
2. Number of Positive Integers: There are 3 positive integers.
3. Sign of Product of Positive Integers: The product of positive integers is always positive. So multiplying 3 positive integers together results in a positive product.
4. Sign of Product of Negative Integers: The sign of the product of an odd number of negative integers is negative. This is because each pair of negative integers results in a positive value (negative x negative = positive), but with an odd number, there will be one negative left unpaired, yielding a negative product overall.
Thus, [tex]\( (- \cdot - = +\)[/tex] for even count repeats and leaves an odd one, leading to [tex]\(-) \)[/tex].
5. Final Product Sign: Combining the 3 positive integers (which is positive) and the product of 105 negative integers (which is negative) results in a negative product overall.
Therefore, the sign of the product for part (a) is negative.
### Part (b)
Question: What will be the sign if you multiply 64 negative integers and 1 positive integer?
Step-by-Step Solution:
1. Number of Negative Integers: There are 64 negative integers.
2. Number of Positive Integers: There is 1 positive integer.
3. Sign of Product of Positive Integers: The product of positive integers is always positive. Here, multiplying 1 positive integer results in a positive product.
4. Sign of Product of Negative Integers: The sign of the product of an even number of negative integers is positive. This is because each pair of negative integers results in a positive value (negative x negative = positive), and with an even number, all negative integers can be paired.
Thus, [tex]\( (- \cdot - = +\)[/tex] repeating results in [tex]\(+\)[/tex] consistently in complete pairs.
5. Final Product Sign: Combining the 1 positive integer (which is positive) and the product of 64 negative integers (which is also positive) results in a positive product overall.
Therefore, the sign of the product for part (b) is positive.
### Summary:
(a) The product of 105 negative integers and 3 positive integers is negative.
(b) The product of 64 negative integers and 1 positive integer is positive.
