Question

☺️☺️50 POINTS___________

♣️Q. - IF a AND b ARE TWO ODD POSITIVE INTEGERS SUCH THAT a > b ,THEN PROVE THAT ONE OF THE TWO NUMBERS ab/2 AND a-b/2 IS ODD AND THE OTHER IS EVEN.♣️

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Answer 1

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Answer 2

Let us suppose that the numbers (a+b)/2 and (a-b)/2 are either both odd or both even.

Case1) When (a+b)/2 and (a-b)/2 are odd.

We know that the sum or difference of two odd numbers is even, hence the sum of the numbers (a+b)/2 and (a-b)/2 must be even.

So, (a+b)/2 +(a-b)/2=a must be even which is not correct as we are given that a is odd positive integer.

(If we take the difference, we will get the value as equal to b).

This leads to a contradiction. Hence (a+b)/2 +(a-b)/2 cannot be both odd.

Case 2) When (a+b)/2 and (a-b)/2 are even.

Again, the sum or difference of two even numbers is even.

So, (a+b)/2 +(a-b)/2=a must be even, which is not correct as we are given that a is odd positive integer. (If we take the difference, we will get value equal to b). This again leads to a contradiction. Hence (a+b)/2 +(a-b)/2 cannot be both even.

So, the given two numbers cannot be both even or both odd. Hence, there is only one possibility that one out of a+b/2 and a-b/2 is odd and the other is even.