Question

Critical Thinking: 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 is even; y is odd.

Short answer

If x is even integer and y is odd integer, then their sum is always odd.

Step-by-step solution

– 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.

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 even integers is always even. 

The sum of one even integer and one odd integer is always odd.

Explanation:

Even numbers are always multiples of 2.

If x is even and y is odd integers then there exists integers a, b such that x = 2a and y = 2b + 1

x + y = 2a + 2b + 1 = 2(a + b) + 1 . We can write x + y is an odd integers.

Example: x = 14 and y = 19, then x + y = 14 + 19 = 33.

14 is an even integer, 19 is an odd integer but their sum 33 is always an odd integer.

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