If a is an odd number, b is an even number, and c is an odd number, then the following expressions will always be equivalent to an odd number:
a + c
a - c
c + a
c - a
a + c
a - c
c + a
c - a
In all of these expressions, the sum or difference of two odd numbers is always an odd number, so the result will always be an odd number.
On the other hand, the following expressions will always be equivalent to an even number:
a + b
a - b
b + a
b - a
a + b
a - b
b + a
b - a
In all of these expressions, the sum or difference of an odd number and an even number is always an even number, so the result will always be an even number.