Answer: B) Odd
Step-by-step explanation:
To prove that the sum of an even integer and an odd integer is an odd integer, let's use some basic principles of integer properties.
Let n1 be an even integer, and let n2 be an odd integer.
By definition:
An even integer can be expressed as n1 = 2k, where k is some integer.
An odd integer can be expressed as n2 = 2m + 1, where m is some integer.
Now, let's calculate the sum n1 + n2:
n1 + n2 = (2k) + (2m + 1)
Now, we can factor out the common factor of 2:
n1 + n2 = 2(k + m) + 1
Notice that the expression inside the parentheses, (k + m), is also an integer because the sum of two integers is always an integer.
So, we have:
n1 + n2 = 2(some integer) + 1
Since we have an expression in the form of 2(some integer) + 1, it is clear that n1 + n2 is an odd integer because it can be expressed as 2 times some integer plus 1, which is the definition of an odd integer.
So, the answer is:
B) Odd