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Home » 1. 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 is even.
1. 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 is even.
CBSE | Class 10 | Maths | RD Sharma | Real Numbers
1. 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 is even.

Solution: Real numbers are simply the combination of rational and irrational numbers, in the number system. In general, all the arithmetic operations can be performed on these numbers and they can be represented in the number line

We know that any odd positive integer is of the form 4q+1 or,4q+3 for some whole number q.

Now that it’s given a > b

So, we can choose a= 4q+3 and b= 4q+1.

∴ (a+b)/2=[(4q+3)+(4q+1)]/2

⇒ (a+b)/2=(8q+4)/2

⇒ (a+b)/2=4q+2=2(2q+1) which is clearly an even number.

Now, doing (a-b)/2

⇒ (a-b)/2=[(4q+3)-(4q+1)]/2

⇒ (a-b)/2=(4q+3-4q-1)/2

⇒ (a-b)/2=(2)/2

⇒ (a-b)/2=1 which is an odd number.

Hence, one of the two numbers (a+b)/2 and (a-b)/2 is odd and the other is even.

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