Question

When is the product not greater than both of its factors?

Answer 1

The product of two numbers is not greater than both of its factors when at least one of the factors is either zero or one. Here’s the reasoning:

1. **If one of the factors is zero:**
\[
a \times 0 = 0
\]
In this case, the product (0) is equal to one of the factors (0) and therefore is not greater than that factor.

2. **If one of the factors is one:**
\[
a \times 1 = a
\]
Here, if \( a \) is greater than 1, then the product \( a \) is greater than one of its factors (1). However, if \( a \) is less than or equal to 1 (i.e., \( a = 1 \) or \( a < 1 \)), then the product \( a \) is not greater than that factor (1).

3. **Negative Factors:**
If both factors are negative (for example, \( a \) and \( b \) are both negative):
\[
(-a) \cdot (-b) = ab
\]
Here, \( ab \) is positive, but both \( -a \) and \( -b \) are less than their corresponding products, which can be counterintuitive.

In summary, the product of two numbers \( a \) and \( b \) will not be greater than both of its factors when:
- Either \( a \) or \( b \) (or both) equals 0.
- At least one of the factors is 1 and the other factor is less than or equal to 1.
- Both factors are negative numbers.