Question

Show that the sum of two even integers is even.

Answer 1

The sum of two even integers is always even.

Step-by-step explanation:

In mathematics, an even integer is defined as any integer that is divisible by 2. Let's denote two even integers as
`\(2n_1\)` and
`\(2n_2\)`, where
`\(n_1\)` and
`\(n_2\)` are integers. The sum of these two even integers is given by:

`\[2n_1 + 2n_2\]`

Factoring out the common factor of 2, we get:

`\[2(n_1 + n_2)\]`

Since
`\(n_1\)` and
`\(n_2\)` are integers, their sum
`\(n_1 + n_2\)` is also an integer. Therefore, we can represent it as another integer
`\(n\)`, making the expression:

`\[2n\]`

This final expression clearly shows that the sum is an even integer since it is a multiple of 2. Thus, we have proved that the sum of two even integers is indeed even.

In essence, when you add two even integers, the result is always a multiple of 2. This property holds true regardless of the specific values of the even integers. Whether you add two, four, or any number of even integers, the sum will always be an even number. This mathematical principle provides a clear and concise explanation for the statement that the sum of two even integers is always even.