Question

For any non-zero integers an and b, a x b and a/b always have the same sign. Correct or incorrect

Answer 1

The statement is incorrect.

To understand why, let's analyze the signs of the products and quotients of two non-zero integers \( a \) and \( b \).

1. **Product \( a \times b \)**:
- If both \( a \) and \( b \) are positive, then \( a \times b \) is positive.
- If both \( a \) and \( b \) are negative, then \( a \times b \) is also positive (since a negative times a negative is positive).
- If one is positive and the other is negative, then \( a \times b \) is negative.

2. **Quotient \( a / b \)**:
- If both \( a \) and \( b \) are positive, then \( a / b \) is positive.
- If both \( a \) and \( b \) are negative, then \( a / b \) is also positive (since dividing a negative by a negative yields a positive).
- If one is positive and the other is negative, then \( a / b \) is negative.

From this analysis, we can conclude that:

- \( a \times b \) and \( a / b \) are both positive when both integers have the same sign.
- \( a \times b \) and \( a / b \) are both negative when one integer is positive and the other is negative.

Thus, \( a \times b \) and \( a / b \) have the same sign if both integers have the same sign, but when their signs differ, the product and quotient also differ in sign.

In summary, both the product and the quotient do have the same sign when both integers are either positive or negative, but in mixed sign cases, they do not share the same sign. Therefore, the claim is **not universally true**.