Question

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

Answer 1

Let a = 2n-1 and b = 2n-5 be two odd numbers for any n, and a>b.

 

Then, (a + b) /2 = (2n – 1 + 2n – 5)/2 = (4n – 6) = 2n – 3.

 

If 2n-1 and 2n-5 are odd numbers, then 2n-3 is also an odd number. Thus (a + b)/2 is an odd number, for any n.

 

Similarly, (a – b) / 2 = (2n – 1 – 2n + 5) / 2 = 4 / 2 = 2.

 

And 2 is an even number. Thus, (a – b)/2 is an even number.

 

Thus, proved that for any two odd numbers, a and b, (a + b)/2 is odd and (a – b)/2 is even.