Question

Give a direct proof that the sum an even integer and an odd integer is odd.

Answer 1

The sum of an even integer and an odd integer is odd.

Step-by-step explanation:

To give a direct proof that the sum of an even integer and an odd integer is odd, we can use the definition of even and odd numbers. Let's say we have an even number represented by 2n and an odd number represented by 2m+1, where n and m are integers. The sum of these two numbers is 2n + (2m+1) = 2n + 2m + 1 = 2(n+m) + 1.

Since n+m is an integer, let's say k, we can rewrite the sum as 2k + 1, which is an odd number. Therefore, the sum of an even integer and an odd integer is odd.

Answer 2

Even integer: 2 x a, where a is a natural number without 0;
Odd integer : 2 x b + 1 , where b is a natural number;
So, 2 x a + 2 x b+ 1 = 2 x ( a + b ) + 1, which is an odd integer.