Chapter 2.7, Problem 12OE

To determine

To verify: $m-n,m+n,mn$ are odd or even

Answer to Problem 12OE

  $m+n,m-n$ are odd

  $mn$ is even

Explanation of Solution

Given information: $m$ is odd and $n$ is even

Concept used: A number of the form $2p+1$ is odd and number of the form $2p$ is even where $p$ any integer is.

As $m$ is odd and $n$ is even, take $m=2k+1,n=2l$ where $k,l$ are integers.

First, we will check whether $m+n$ is odd or even

Put $m=2k+1,n=2l$ in $m+n$

  $\begin{array}{l}m+n=(2k+1)+2l\\ =(2k+2l)+1\\ =2(k+l)+1\end{array}$

Take $k+l=p$

So,

  $m+n=2p+1$ which is an odd integer

Hence, $m+n$ is an odd integer.

Next, we will check whether $m-n$ is odd or even.

Put $m=2k+1,n=2l$ in $m-n$

  $\begin{array}{l}m-n=(2k+1)-2l\\ =(2k-2l)+1\\ =2(k-l)+1\end{array}$

Take $k-l=p$

So,

  $m-n=2p+1$ which is an odd integer

Hence, $m-n$ is an odd integer.

Now, check whether $mn$ is odd or even

Put $m=2k+1,n=2l$ in $mn$

  $\begin{array}{l}mn=(2k+1)(2l)\\ =2k(2l)+1(2l)\\ =4kl+2l\\ =2(2kl+l)\end{array}$

Take $p=2kl+l$ .

So,

  $mn=2p$ which is an even integer.

Hence, $mn$ is an even integer.