Final answer:
When adding an even integer to an odd integer, the sum is always an odd number. This occurs because the sum of an even and an odd number ultimately results in an even number plus 1, which is always odd.
Step-by-step explanation:
When adding an even integer to an odd integer, the sum is always an odd number. This is because an even number can be expressed as 2n, where n is an integer, and an odd number can be expressed as 2m+1, where m is also an integer. When we add these together (2n + (2m+1)), we can group the even parts together and the odd part stands alone, resulting in (2n + 2m) + 1, which is an even number plus 1, thus always odd.
For example, if we take 4 (an even number) and add it to 3 (an odd number), 4 + 3 equals 7, which is an odd number. Another example would be adding 10 (even) and 5 (odd), which results in 15 (odd). So, the rule for adding an even and an odd integer is that the result will always be an odd number.