Question

If m and n are both odd integers, which of the following must also be an odd integer?
1. 2m +n
2. (m +n)^2
3. mn

Answer 1

To determine whether each expression is odd when \( m \) and \( n \) are both odd integers, let's examine each case.

1. **For \( 2m + n \)**:
- Since \( m \) is odd, \( 2m \) is even (as multiplying an odd integer by 2 results in an even integer).
- \( n \) is odd.
- The sum of an even integer and an odd integer is odd.
- Therefore, \( 2m + n \) is odd.

2. **For \( (m + n)^2 \)**:
- The sum \( m + n \) is the sum of two odd integers, which is even (the sum of two odd numbers is always even).
- Squaring an even integer results in an even integer.
- Therefore, \( (m + n)^2 \) is even.

3. **For \( mn \)**:
- The product of two odd integers is always odd.
- Therefore, \( mn \) is odd.

In summary:
- \( 2m + n \) is odd.
- \( (m + n)^2 \) is even.
- \( mn \) is odd.

The expressions that must also be odd are:
- \( 2m + n \)
- \( mn \)

So the expressions that must be odd are:
1. **True**: \( 2m + n \) is odd.
2. **False**: \( (m + n)^2 \) is even.
3. **True**: \( mn \) is odd.

Thus, the final answer is that the expressions \( 2m + n \) and \( mn \) are odd (while \( (m + n)^2 \) is not).