Given:a and b are two positive integers where a>b
To prove: a+b/2 and a-b/2 is odd and the other is even
We know that any positive integer is of the form q,2q+1
let a=2q+1 and b=2m+1 where q and m are some whole nos.
=q+m+1 which is a +ve integr
now,on substituting the values of a and b in a-b/2
we get a-b/2=q-m
Thus,a-b/2 is a +ve integer
Now we've to prove that a+b/2 and a-b/2 is either odd or even
assume that a+b/2 - a-b/2
=a+b-a+b/2=2b/2=b which is an odd +ve integer
Also we proved that a+b/2 and a-b/2 are +ve integers
We know that the diffrence of 2 +ve intg. is an odd no. if one of them is odd and the other is even(also,diff. between two odd and two even integ. is even)
Hence it is proved
Hope it helps!!!!