# 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.

1
by Avishek

Log in to add a comment

by Avishek

Log in to add a comment

The Brainliest Answer!

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