This is a class 10 question. If a and b are two odd positive integers such that ab, then prove that one of the two numbers a+b/2 and a-b/2 is odd and the other is even.

Question

This is a class 10 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/2 is odd and the other is even.

Answer

Let a=2q+3 and b=2q+1 be two odd positive integers such that a>b.Now, a+b/2=2q+3+2q+1/2=4q+4/2=2q+2 which is an even number.and, a-b/2=2q+3-(2q+1)/2=2q+3-2q-1/2=2/2=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 for any two positive odd integers.

Answer

Answer

Let us consider that (a+b)/2 and (a-b)/2 are both odd and both even In the first case, let them be 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 value equal to b). This leads to a contradiction. Hence (a+b)/2 +(a-b)/2 cannot be both odd. In the second case, let the numbers (a+b)/2 +(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 leads to a contradiction. Hence (a+b)/2 +(a-b)/2 cannot be both even. So, the two numbers in question cannot be both even or both odd. Hence they have only one possibility left – one is even and the other is odd

Answer

Understanding Odd Positive Integers
Let $ a $ and $ b $ be two odd positive integers where $ a > b $. An odd integer can be expressed in the form:
- $ a = 2m + 1 $
- $ b = 2n + 1 $
where $ m $ and $ n $ are non-negative integers.
Calculating $ a + b $ and $ a - b $
Now, we can calculate $ a + b $ and $ a - b $:
- $ a + b = (2m + 1) + (2n + 1) = 2(m + n + 1) $
- $ a - b = (2m + 1) - (2n + 1) = 2(m - n) $
Finding $ \frac{a + b}{2} $ and $ \frac{a - b}{2} $
Next, we evaluate the expressions $ \frac{a + b}{2} $ and $ \frac{a - b}{2} $:
- $ \frac{a + b}{2} = \frac{2(m + n + 1)}{2} = m + n + 1 $ (which is an integer)
- $ \frac{a - b}{2} = \frac{2(m - n)}{2} = m - n $ (which is also an integer)
Determining Parity
- Since $ m + n + 1 $ is the sum of two integers $ m + n $ (even if both $ m $ and $ n $ are even or odd) plus 1, it is odd.
- $ m - n $ can either be even or odd, depending on the values of $ m $ and $ n $.
Conclusion
Therefore, we conclude that:
- One number $ \frac{a + b}{2} $ is odd.
- The other number $ \frac{a - b}{2} $ is even or vice versa.
This means one of the expressions is odd and the other is even, as required.

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