Odd numbers becoming even numbers but even numbers not becoming odd numbers

Question

When you multiply an odd number by an even number you get an even number, right? For example, $19 \cdot 2$ that is $38$ (an even number). But when you try applying that logic to even numbers it will fail.
For example, $2 \cdot 19$ again, which is $38$, and $38$ is an even number, so multiplying an even number by an odd number does not make it an odd number.
But why? If one is like that why should not the other be like it too?

Answer 1

You can figure this out. An even number of cookies means you can divide the cookies evenly between two people without breaking any cookies. An odd number of cookies means there is an odd cookie left over that must be broken in half.

Suppose you have bags of cookies, each bag containing the same number of cookies.

If the number of bags is even, you can certainly divide the cookies evenly between two people: just give each person the an equal number of bags. This divides the cookies evenly, no matter how many cookies are in each bag.

If the number of cookies in each bag is even, you can divide the cookies between two people: just give each person the same number of cookies from each bag. This divides the cookies evenly, no matter how many bags there are.

(If both numbers are even, you can do it either way, they both work.)

But if the number of bags is odd, and the number of cookies in each bag is odd, then you can't divide the cookies evenly. You can give half the bags to each person, and then there will be one bag left over. And when you try to divide the odd number of cookies in the last bag, there will be one cookie left over that must be broken.

Answer 2

A number, if not prime, is "composed" of two or more multiplicators. Thus an even number includes one (or more) $2$. No matter if you multiply an odd with an even number or an even numer with an odd, the result will include the $2$ now, what makes it an even number.

Addendum: those multiplicators making up a number may be split as small as possible, down to primes. Those are called the prime factors of a number. Fun fact -- $2$ is the only even prime. Prime numbers consist of only one multiplicator, or prime factor (besides the $1$, which is not regarded as a prime number).

Answer 3

• "Even" means "multiple of 2".
• "Odd" means "not multiple of 2".

Those are the definitions of even and odd.

The fact that even numbers end in 0,2,4,6,8 and odd numbers end in 1,3,5,7,9, when written in decimal, is a much less fundamental and much less enlightening property of even and odd numbers.

So when you write "For example, 19⋅2 that is 38 (an even number)", the reasoning seems backward: it should be obvious that 19⋅2 is a multiple of 2, and you shouldn't have to perform the calculation to know that 19⋅2 is a multiple of 2. As for 38, you can check that it is a multiple of 2 by decomposing it into the multiplication 19⋅2, or by checking the useful but mysterious property that if its last digit is an 8, then it must be a multiple of 2.

If a number if a multiple of 2, you can multiply it more and more if you want, it will still be a multiple of 2. This is why multiplying an even number, i.e. a multiple of 2, by any number, no matter whether the other number is even or odd, will always result in a multiple of 2.

You can say that the property of being a multiple of 2 is "absorbing" for multiplication. The more you multiply things together, the more opportunities to incorporate a 2 in the multiplication. Once a 2 is in the multiplication, multiplying more and more will not remove this 2.

The fact that when multiplying an odd number by an odd number, the result is still odd, is a bit more subtle. It comes down to the fact that number 2 is "unbreakable", i.e. you cannot make a 2 appear by multiplying two things that do not contain 2.

2 is not the only number with this unbreakable property.

2, 3, 5, 7, 11, 13, 17, and many more, share the unbreakable property: if you multiply two numbers which are not multiples of 3, then the result is not a multiple of 3 either.

But most numbers do not have this unbreakable property. For instance, 6 doesn't have the property. You can get a multiple of 6 by multiplying things that are not multiples of 6. For instance, 10 is not a multiple of 6, and 15 is not a multiple of 6, and yet 10⋅15 = 150 = 6⋅25 is a multiple of 6. This is because 6 can be broken into 2⋅3, so when you multiply 10, which is a multiple of 2, by 15, which is a multiple of 3, you can get a multiple of 6.

Answer 4

In mathematics (in fact, anywhere) you can't reliably predict what will happen using just intuition and analogy. Those tools can suggest what might be true, but then you have to check the mathematical reality.

In your example, your intuition suggested to you that even $\times$ odd should somehow be the opposite of odd $\times$ even. That happens to be false, because in multiplication the order of the factors doesn't matter. So the fact that the answers are the same, not opposite, is not something special that needs special explanation.

There are situations where the order in which you do things matters. Putting on your socks before your shoes is not the same as putting on your shoes first.

There are mathematical situations where order matters too. If you start from New York and travel north for 1000 miles and then east for 1000 miles you will end up in a different place than if you went east first and then north.

Answer 5

While it is true that one cannot multiply an even integer by another integer and get an odd integer ($(2 \cdot a \cdot b \cdot \dots) \cdot c $ still has that pesky two), one could, of course, multiply an even number by $\frac{1}{2}$ until all of those 2's in the prime factorization are gone. This remains true if you extend your idea of evenness to "divisibility by $n$" where $n$ is any prime number, i.e. $(n \cdot a \cdot b \cdot \dots) \cdot c$ still contains that pesky $n$. Of course, if you accidentally multiplied an odd number by $\frac{1}{2}$, you would move yourself out of the realm of integers and into the realm of rationals.

Integer multiplication with evens and odds actually forms a truth table - an AND gate, if you accept that odds are 1 (an odd number which we map to "true") and evens are 0 (an even number that maps to false) (or a NAND gate if you don't).

Your conjecture - that an odd times an even gives an even, but an even times an odd gives an odd would require that integer multiplication no longer commutes, i.e. that $e \cdot o \neq o \cdot e$ (where $e$ and $o$ are even and odd integers). This is clearly false for all real and complex numbers, but even toddlers know that putting on socks before shoes is different than putting on shoes and then socks.

In general, multiplying matrices is non-commutative as is multiplication for quaternions and octonians. Unfortunately, the concept of even and odd for these objects is significantly muddled. With matrices, apparently somebody has defined and made some conclusions about odd and even matrices, but they seem both obscure and beyond the scope of your question.

With Quaternions (octonians and sedonians), one could probably use Lipschitz integers (like Gaussian integers for complex numbers) to figure out something, but quaternions have 4 components, so instead of having 4 categories of even and odd as do complex numbers with only 2 components (evens, even-odd, odd-even and odds), you'd end up with $2^4 = 16$ categories. This too is beyond the scope of your question. The point of all of this diversion is that people have examined things that behave like you want, but they were significantly more complicated than your question leads one to believe you want the answer to be. If you only ever multiply by integers, every factor you ever multiplied your number by remains until you incorporate division.