RD Sharma Solutions For Class 10 Math Chapter 1 Real Numbers Exercise 1.1

 Q1. If a and b are two odd positive integers such that a > b, then prove that one of the two numbers $\frac{a+b}{2}$ and $\frac{a-b}{2}$ is odd and the other is even. 

Solution: 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. 

∴ $\frac{a+b}{2}=\frac{\left[ (4q+3)+(4q+1) \right]}{2}$  

⇒ $\Rightarrow \frac{a+b}{2}=\frac{8q+4}{2}$(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 

⇒ $\Rightarrow \frac{a-b}{2}=\frac{\left[ (4q+3)-(4q+1) \right]}{2}$  

⇒ (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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