Why when you add odd and even numbers the sum is odd?

Answer 1

Factors can be listed in pairs. Square numbers will have one pair that is the same number. When put into a list, that number will only be listed once and will result in an odd number of factors.

Answer 2

An odd number is defined as a number that leaves a remainder of 1 when divided by 2.

So, if a is an odd number then there exists an integer x such that a = 2x+1 where x belongs to N, the set of integers.

Also, an even number is defined as a number that leaves a remainder of 0 when divided by 2.

So, if b is an even number then there exists an integer y such that b = 2y where y is in N.

Then an odd number + an even number = a + b = 2x+1+2y

=2x+2y+1 (by commutativity)

=2(x+y)+1 (by distributivity, in reverse)

Now x is in N and y is in N means that x+y is in N, say z (closure of N under addition)

Thus a+b = 2z+1 which, by the definition of odd numbers, is odd.