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 you reasoning.

x and y are even.

Short answer

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

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 two odd integers is always even.

Explanation:

Even numbers are always multiples of 2.

If x and y are even integers then there exists integers a, b such that x = 2a and y = 2b.

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

Example: x = 14 and y = 18, then x + y = 14 + 18 = 32.

14, 18 and 32 all are even number.

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