Question

Let a and be any positive integer where a > b. Show that either of (a+b)/2 and (a-b)/2 is an even number.

Answer 1

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 
proof:
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.

---> a+b/2=(2q+1)+(2m+1)/2
              = 2q+2m+2/2
              = 2(q+m+1)/2
              =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
GIVEN, a>b
         therefore, 2q+1>2m+1
                         2q>2m
                           q>m
therefore,a-b/2=(q-m)>0
 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!!!!