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 questionCase $$I:$$ $$\dfrac {a+b}{2} =\dfrac {2n+3+2n+1}{2}$$$$=\dfrac {4n+4}{2}$$$$=2n+2=2(n+1)$$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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