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/2is odd and the other is even

Answer 1

Let a and b are any two odd positive integers.

Hence and where m and n are whole numbers.

Consider

Therefore is a positive integer.

Now,

But given a > b









Hence is also a positive integer

Now we have to prove that of the numbers and is odd and another is even number.

Consider,

which is an odd positive integer à (1)

It is already proved that and are positive integers à (2)

Recall that the difference between an odd number and even number is always an odd number.

Hence from (1) and (2), we can conclude that one of the integers and is even and other is odd.

Recommend(0) Comment (0) more_horiz
person
Kishore , Student

Member since Dec 10 2008


Solution:



Let a and b are any two odd positive integers.

Hence and where m and n are whole numbers.

Consider

Therefore is a positive integer.

Now,

But given a > b









Hence is also a positive integer

Now we have to prove that of the numbers and is odd and another is even number.

Consider,

which is an odd positive integer à (1)

It is already proved that and are positive integers à (2)

Recall that the difference between an odd number and even number is always an odd number.

Hence from (1) and (2), we can conclude that one of the integers