Question

Using the given information and the fact that x and y are integers, tell whether the sum x + y is even or odd. Explain your reasoning.

x and y are odd.

Short answer

If x and y are two odd integers, then their sum is always an even integer.

Step-by-step solution

Step 1. Definition of even and odd numbers.

An even number is a number that can be divided into two equal groups. Even numbers end in 2, 4, 6, 8 and 0 regardless of how many digits they have.

An odd number is a number that cannot be divided into two equal groups.

Examples of odd numbers are 3 , 5 , 7 , 9 , 11, ......

Rules for addition and subtraction of two even integers or odd integers 

1. An even number can only be formed by the sum of either 2 odd numbers (odd + odd = even), or 2 even numbers (even + even = even).

2. An odd number can only be formed by the sum of an odd and even number (odd + even = odd, or even + odd = odd).

Explanation for sum of two odd integers is even. 

The sum of two odd integers is always even.

Explanation:

Even numbers are always multiples of 2.

If x and y are two odd integers then there exists integers a, b such that x = 2a + 1 and y = 2b + 1.

x + y = 2a +1+ 2b + 1 = 2(a + b + 1). We can write x + y is always even.

Example: x = 15 and y = 21, then x + y = 15 + 21 = 36.

15, 21 are odd but the sum 36 is an even number.

Conclusion: Sum of two odd numbers is always an even number.