Question

If $a$ and $b$ are two odd positive integers such that $a>b,$ then prove that one of the two numbers $\frac{a+b}{2}$ and $\frac{a-b}{2}$ is odd and the other is even.

Answer 1

Since a and b are odd numbers , we can write $a=2n+1$ and $b=2n^{\prime }+1$

So $\frac{a+b}{2}=$ $n+n^{\prime }+1$ and $\frac{a-b}{2}=$ $n-n"$

If $n+n^{\prime }$ is odd then $n-n^{\prime }$ must also be odd and If $n+n^{\prime }$ is even then $n-n^{\prime }$ must also be even .

Thus we can also say that If $n+n^{\prime }+1$ is odd then $n-n^{\prime }$ must be even and

If $n+n^{\prime }+1$ is even then $n-n^{\prime }$ must be odd .

So it is proved that one of $\frac{a+b}{2}$ and $\frac{a-b}{2}$ is odd and other is even