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Question

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


Solution

We have
$$a$$ and $$b$$ are two odd positive integers such that $$a$$ & $$b$$
but we know that odd numbers are in the form of $$2n+1$$ and $$2n+3$$ where $$n$$ is integer.
so, $$a=2n+3,\ b=2n+1,\ n\in 1$$
Given $$\Rightarrow \ a > b$$
now, According to given question

Case $$I:$$ 
$$\dfrac {a+b}{2} =\dfrac {2n+3+2n+1}{2}$$
$$=\dfrac {4n+4}{2}$$
$$=2n+2=2(n+1)$$
put let $$m=2n+1$$ then,
$$\dfrac {a+b}{2}=2m\ \Rightarrow $$ even number.

Case $$II:$$
$$\dfrac {a-b}{2}=\dfrac {2n+3-2n-1}{2}$$
$$\dfrac {2}{2}=1\ \Rightarrow $$ odd number.
Hence we can see that, one is odd and other is even.
This is required solutions.

Mathematics

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