Question

Find the truth value of a statement “The sum of any two odd numbers is an odd number “.
\[\begin{align}
  & \text{(A) True} \\
 & \text{(B) False} \\
 & (\text{C) Neither True nor false} \\
 & \text{(D) Cannot be determined} \\
\end{align}\]

Answer 1

Hint: First of all, we assume odd numbers in general form. Now let us assume the sum of those assumed odd numbers as S. Now we will find the value of S. Now we should check that the obtained number is a multiple of 2 or not. Because we know that a number is multiple of 2, then the number is said to be even. So, we should check whether S is a multiple of 2 or not. After checking we will make a concluding statement. Now by comparing the concluding statement with the given statement, we can say whether a statement is true or false.

Complete step-by-step answer:
Before solving the question, we should know that the general form of an odd number is (2n+1) where n is an integer.
Let us assume two odd numbers. Let the first odd number is \[(2p+1)\] and the second odd number is \[(2q+1)\], where p and q are integers.
Let us assume the sum of two odd numbers as S.
\[\begin{align}
  & \Rightarrow S=(2p+1)+(2q+1) \\
 & \Rightarrow S=2p+2q+2 \\
 & \Rightarrow S=2(p+q+1)......(1) \\
\end{align}\]
As we know that p, q are integers, so \[(p+q+1)\] is also an integer.
We know that any number which is a multiple of 2 is definitely an even number.
From equation (1), we get that S is a product of 2 and \[(p+q+1)\].
So, we can say that S is a multiple of 2.
So, we can say that S is an even number.
Hence, we can conclude that the sum of any two odd numbers is always even.
But the given statement is “The sum of any two odd numbers is an odd number”.
By comparing the result with the given statement, we can say that the given statement is false.
Hence, option B is correct.

Note: This problem can be solved in an alternative method also. We know that the general form of an odd number is (2n-1), where n is a whole number.
Let us assume two odd numbers. Let the first odd number is \[(2p-1)\] and the second odd number is \[(2q-1)\], where p and q are integers.
Let us assume the sum of two odd numbers as S.
\[\begin{align}
  & \Rightarrow S=(2p-1)+(2q-1) \\
 & \Rightarrow S=2p+2q-2 \\
 & \Rightarrow S=2(p+q-1)......(1) \\
\end{align}\]
As we know that p, q are integers, so \[(p+q-1)\] is also an integer.
We know that any number which is a multiple of 2 is definitely an even number.
From equation (1), we get that S is a product of 2 and \[(p+q-1)\].
So, we can say that S is a multiple of 2.
So, we can say that S is an even number.
Hence, we can conclude that the sum of any two odd numbers is always even.
But the given statement is “The sum of any two odd numbers is an odd number”.
By comparing the result with the given statement, we can say that the given statement is false.