Let's analyze each statement one by one to determine whether they are true or false.
### Step-by-Step Solution:
**i. The product of a positive and a negative integer is negative.**
- When you multiply a positive integer (e.g., +3) with a negative integer (e.g., -2), the result is -6.
- Therefore, this statement is **True**.
**ii. The product of two negative integers is a negative integer.**
- When you multiply two negative integers (e.g., -3 and -2), the result is +6.
- Therefore, this statement is **False**.
**iii. The product of three negative integers is a negative integer.**
- When you multiply three negative integers (e.g., -2, -3, and -4), the calculation goes as follows:
- First, multiply the first two: (-2) * (-3) = +6 (positive).
- Then, multiply the result with the third: +6 * (-4) = -24 (negative).
- Therefore, this statement is **True**.
**iv. Every integer when multiplied with -1 gives its multiplicative inverse.**
- The multiplicative inverse of an integer x is defined as 1/x.
- When you multiply an integer (e.g., 5) by -1, you get -5, not 1/5.
- Therefore, this statement is **False**.
**v. Multiplication on integers is commutative.**
- The commutative property states that changing the order of the numbers does not change the product.
- For example, 2 * 3 = 6 and 3 * 2 = 6.
- Therefore, this statement is **True**.
**vi. Multiplication on integers is associative.**
- The associative property states that the way in which numbers are grouped does not change the product.
- For example, (2 * 3) * 4 = 6 * 4 = 24 and 2 * (3 * 4) = 2 * 12 = 24.
- Therefore, this statement is **True**.
**vii. Every nonzero integer has a multiplicative inverse as an integer.**
- The multiplicative inverse of a nonzero integer x is 1/x, which is not an integer unless x = 1 or x = -1.
- For example, the multiplicative inverse of 2 is 1/2, which is not an integer.
- Therefore, this statement is **False**.
### Summary of Statements:
1. True
2. False
3. True
4. False
5. True
6. True
7. False
