Question

For any positive even integer \( p \), every positive odd integer will be of the form:

• A. \( p \)
• B. \( p + 1 \)
• C. \( 2p \)
• D. \( 2p + 1 \)

Hint:

The form \( 2p + 1 \) always gives the next odd integer after an even integer \( p \).

Answer 1

The Correct Option is D

Step 1: Understanding even and odd integers.
An even integer is any integer that can be written in the form \( 2n \), where \( n \) is an integer. A positive odd integer is one that cannot be divided by 2, and can be written in the form \( 2k + 1 \), where \( k \) is an integer.
Step 2: Applying the condition for the even integer \( p \).
Given that \( p \) is a positive even integer, we can express it as \( p = 2m \) where \( m \) is an integer.
Step 3: Forming the positive odd integer.
To find a positive odd integer that is greater than \( p \), we can add 1 to \( p \): \[ 2m + 1 = 2p + 1 \] This is the required odd integer.
Step 4: Conclusion.
Therefore, every positive odd integer will be of the form \( 2p + 1 \).