Does an odd number plus an odd number equal an odd number?

Answer 1

Well, let's see . . . . .

3 + 5 = 8

5 + 15 = 20

7 + 19 = 26

31 + 27 = 58

49 + 11 = 60

It's looking bad for your proposed rule.

This can actually be proven.

Any odd number can be expressed as 2p + 1 where pis any integer.

Let a = 2p + 1 and b = 2q + 1 where p and q are any two integers. Then, by definition, a and b are both odd, so lets add them up.

We know that any two integers added together equals another integer, the question is whether or not that integer is even or odd.

Let m be the integer such that a + b = m.

Substituting, we get (2p + 1) + (2q + 1) = m or 2p + 2q + 2 = m.

Dividing by 2 we get p + q = m/2 - 1.

Since p and q are both integers, m/2 - 1 must also be an integer. Therefore, m must be an even number.

Substituting the expression for p into the expression for a from above we get:

a = 2(m/2 - 1 - q) + 1 = m - 2 - 2q + 1 = m - (2q + 1) = m - b.

Therefore, a + b = m where a and b were both defined to be odd and m was previously shown to be even.

Thus, an odd + an odd = an even.

Q.E.D.