Question

Can you add 3 odd numbers and get an even number?

Answer 1

Hint: This question can be solved by assuming that the odd number is always of the form 2n+1 where n is even and the even number is of the form 2n where n can be even or odd. By assuming this we can come to the final expression and see which form it comes from.

Complete step-by-step solution:
Let the three odd numbers be of the form 2n+1, 2n+3 and 2n+5, where n is any whole number.
Let $S ={\text{Sum of three odd numbers}}$
$ \Rightarrow S = 2n + 1 + 2n + 3 + 2n + 5$
On simplifying,
$ \Rightarrow S = 6n + 9$
It can also be written as
$S = 6n + 8 + 1$
On taking 2 common from the first two terms,
$ \Rightarrow S = 2(3n + 4) + 1$
$\because 3n+4$ may be even or odd depending on the value of n.
But $2(3n+4)$ is always even because on multiplying any number whether even or odd by 2, the final number becomes even.
$\therefore 2(3n+4)$ is always even.
Also on adding any even number by 1 the final number becomes odd because $odd + even = odd$ .
As $2(3n+4)$ is always even and 1 is always odd.
$\because $ S is of the form $odd + even = odd$ .
$\therefore $ S is always odd.
Hence, it can be concluded as the sum of three odd numbers is always odd.

Note: In this question the general form any number must be remembered to solve the question. In this general form the odd number taken is 2n+1 but it can also be taken as 2n-1 the result would be the same. Always try to solve the question step by step so that the wrong step can be figured out quickly. If the calculations are wrong then the final result would not come. The result that can be drawn from here is the sum of three odd numbers is always odd and also the sum of odd and even numbers is always odd and it can also be checked by solving it with the general form of odd number and even number.