Main content

### Course: 6th grade (Eureka Math/EngageNY) > Unit 2

Lesson 4: Topic D: Number theory—thinking logically about multiplicative arithmetic- Divisibility tests for 2, 3, 4, 5, 6, 9, 10
- Recognizing divisibility
- The why of the 3 divisibility rule
- The why of the 9 divisibility rule
- Divisibility tests
- Intro to even and odd numbers
- Greatest common factor examples
- Greatest common factor explained
- Greatest common factor
- Greatest common factor review
- Least common multiple
- Least common multiple: repeating factors
- Least common multiple of three numbers
- Least common multiple
- Least common multiple review
- GCF & LCM word problems
- GCF & LCM word problems

© 2024 Khan AcademyTerms of usePrivacy PolicyCookie Notice

# Intro to even and odd numbers

This video explains the difference between even and odd numbers, and how they interact when added together. Even numbers are multiples of two and can be divided evenly, while odd numbers are not divisible evenly by two. You can easily identify whether a number is even or odd by looking at the ones place.

## Want to join the conversation?

- @1:21, Khan states that 0 is a multiple of two. Is zero not a multiple of every number?(89 votes)
- Yes, zero is a multiple of every number. That means that you get an even number whenever you multiply by zero, because you get zero.

Be careful though! You cannot say it the other way around and be correct. Two is not a multiple of zero. There is no way to multiply by zero and get two as a result.(84 votes)

- Does even and odd only apply to integers?(18 votes)
- In the context of real numbers, the answer to your question is yes -- the descriptions of "even" and "odd" apply only to integers.

A number n is even if and only if n = 2k, where k is an integer. This definition implies that if n is even then n is an integer, because if k is an integer, then 2k will be an integer.

A number n is odd if and only if n = 2k + 1, where k is an integer. This definition implies that if n is odd then n is an integer, because if k is an integer, then 2k + 1 will be an integer.

Example of an even number --

Consider the number 8

8 = 2*4

4 is an integer. Since 8 can be expressed as 2*4, this means 8 is even.

Example of a number that is not even --

Consider the number 2.2

2.2 = 2*(1.1)

1.1 is not an integer. Since 2.2 cannot be expressed as 2k, where k is an integer, this means 2.2 is not even.

Example of an odd number --

Consider the number -7

-7 = 2*(-4) + 1

-4 is an integer. Since 7 can be expressed as 2*(-4) + 1, this means 7 is odd.

Example of a number that is not odd --

Consider the number 13/10

13/10 = 2*(3/20) + 1

3/20 is not an integer. Since 13/10 cannot be expressed as 2k + 1, where k is an integer, this means 13/10 is not odd.

The concept of parity (the property of being even or odd) can apply to other objects in math as well. For instance, certain functions can be described as odd or even.(17 votes)

- How is 0 an even number?(1 vote)
- mathematically, any number that can be divided by two to create another whole number is even. Zero passes this test because if you halve zero you get zero and Every integer is either of the form (2 × ▢) + 0 or (2 × ▢) + 1; the former numbers are even and the latter are odd. For example, 1 is odd because 1 = (2 × 0) + 1, and 0 is even because 0 = (2 × 0) + 0.

Wanna get some information about Even numbers?: https://www.khanacademy.org/math/cc-2nd-grade-math/cc-2nd-place-value/x3184e0ec:even-and-odd-numbers/v/understand-even-and-odd-numbers-visually.(9 votes)

- Can't an Even + Odd become a even too?(5 votes)
- No, an even number added to an odd number is always an odd number. Try counting out in small numbers by yourself, and you'll notice the result.(3 votes)

- Sal said zero is a multiple of two but, it's also a multiple of 1 so, doesn't that make it odd and even?(5 votes)
- Yes, zero is a multiple of one, but it's considered even because (according to math's principles) all even numbers are two more than the last, and since one is one more than zero, it is not even, so you are half-right,Arbaaz.(4 votes)

- Will 190,398,928 be even?(4 votes)
- Just take a look at the last number. If it is 2, 4, 6, 8, or 0 the number is even. If it is 1, 3, 5, 7, or 9, then the number is odd. So in this case, the number is 8 so the number is even.(5 votes)

- Is 2.3 an even or an odd number? Is 3.2 even?(3 votes)
- Good question! We only talk about even and odd for whole numbers. Numbers like 2.3 and 3.2 are not whole numbers, they're decimals. So, they're neither even nor odd.(5 votes)

- is a negative number even if its multiple by 2 like -68?(3 votes)
- Yes, negative numbers can be even, because they are multiples of 2. Zero is also an even number.(2 votes)

- I have a questions that troubles me. You have made a reasonable case for claiming that the number '0' is even. You present your proof in in a rather mechanical way. On a deeper level however, that seems wrong. the symbol '0' represents 'Nothing'. How can 'nothing' be either even or odd? It is the absence of anything, if you have 0 apples, do you have an even or an odd number of apples? the answer is obvious, you have neither an even nor an odd number of apples, you have NO apples.

On the surface '0' can be classified as an even number but on a conceptual level I don't see how it can be even or odd.

what am I not understanding? Jim(3 votes)- Well, it isn't really the case like the number 1, where it is neither prime nor composite, but rather because 0 is simply divisible by 2 (the definition of "even") and that it is also in between 2 odd numbers, so it just comes down to logic.(2 votes)

- Are Irrational numbers odd or even?(2 votes)
- Neither, because they never end.

pi = 3.1415....etc. If we cut it off at 4 and called it even then not only would it (technically )not be pi, but we'd also be ignoring the rest of the digits in pi.

Same if we cut it off at 3.14159 and called it odd. It's wrong, and we'd be ignoring the other numbers that came after it.

We could say that pi is indeed a rational number because we just use 22/7 and say that it's pi as represented by a fraction and therefore rational, but fractions are also neither even or odd so that doesn't help either.

Conclusion: irrational numbers can't ever be considered odd or even.(3 votes)

## Video transcript

- What I'm going to introduce
you to in this video is a way to classify numbers
as either being even, or being odd. So what does it mean to be even? Well an even number is one where if you had that many doughnuts, you could split it evenly
between two people. So even numbers are numbers
that are multiples of two. So if you have two doughnuts,
you could split them. If you had two people, you could give one
doughnut to each person. Four doughnuts, you could split that
evenly with two people. They could each get two doughnuts. And then we could keep going. All of the multiples of two, these are even numbers. 8, 10, 12, and of course
we could keep going on, and on, and on, and on. An easy way to spot an even number is that it's ones place
is going to be even. So, for example, the
number 32 is an even number because in the ones place you have a two. The number 5,977,354, 5,977,354, well that's an even number,
because the ones place is even. Now there's one number
that's an interesting one, why some people sometimes say, "Well is this one really even?" And that's the number zero. And the number zero is even because it is a multiple of two. How is it a multiple of two? Well, zero times two is equal to zero. So zero is a multiple of two, and so zero, for sure, is an even number. And then that actually makes our looking at the ones
place idea hold up. Because a number like... a number like 150 is an even number, and we can look at the
ones place of it and see, we have a zero there. And a zero is an even number. So if, in the ones place,
you have an even number, you are looking, the
whole thing is going to be an even number. So what are odd numbers? Well one way to think about them is they're the numbers that aren't even. So not multiples of two. Not multiples... not multiples of two. So what are some examples of odd numbers? Well one, three, five, seven, nine, and of course you can
go on, and on, and on. And just like we could
look at the ones place to spot an even number, you
can also look at the ones place to spot an odd number. The number 59, well I have a nine over
here in the ones place. Nine is odd, so this is
going to be an odd number. The number 1,441 has a one in the ones place. That is an odd number. So this whole thing is going to be odd. And so what's another way of... why is it called odd? Well, it'd be hard to
split 1,441 doughnuts. You can't split it
evenly between two people without breaking up a doughnut. If you wanted to leave
the doughnuts whole, one person would have to get one extra doughnut than the other person. You can't split it evenly. The easiest way I think about odd, it's not a multiple of two, it's not even. So another way, it is not... it is not even. So now that we know what an
odd or an even number is, let's think about what
happens when we operate on odd or even numbers. So let's think, in particular, actually let me move over
to the side right over here. Let me move over.... right over here. Let's think, whoops. Fell off the screen. Let's think about what happens
if I take an even number, an even number, plus another even number. Is this number, the sum, is
it going to be even or odd? Well you could take some examples. If I say two plus six, that is going to be equal to eight. Well eight is an even number. If I say 14 plus four, well that equals 18. Once again, eight in the ones place. This is an even number. If I say 150... 156 plus... plus 100... 100 and... why don't we do a simpler number, plus four, that is equal to 160. Zero in the ones place,
this also is an even number. So it looks like all the
examples that I've done so far, when I add an even to an
even, I get an even number. And I encourage you to
keep trying this out. I encourage you to keep trying, pick some even number,
then another even number, add them together and you'll
see that you keep getting even numbers. And it makes sense. Because if you had one number
that's a multiple of two, and you add it to another
number that's a multiple of two, it makes sense that the sum is going to be a multiple of two. Now what happens if you
add an odd and an odd? What happens if you add an odd number plus another odd number? Well let's try. What is one plus three? Well that's equal to four. You actually get an even number. You add two odd numbers,
you get an even number. Well maybe that was
just that special case. What if I add 15 plus seven? These are both odd numbers, when I add them together I get 22. I get another even number. This is interesting. If I take, let me say 19 plus 3. Actually that gets us 22 again. We have an even number. This is interesting. What about, what about, let's see... 23 plus 5, well that gets us to 28. Once again, get an even number. So the pattern that
seems to be forming here is that if I have an even
plus an even I get an even, but also if I have an odd plus an odd, then I also get an even number. And once again, I encourage you to try out as many numbers as you can
to see if this pattern holds. And you will see that it is true. An odd plus an odd number is an even. Now let's think about
one last combination. What about an even, what about an even plus an odd? An even plus an odd. So let's say I have two plus one. Two plus one, well what
is that going to be? Well that's going to be equal to three. That's going to be an odd number. Well what about, what
about four plus three? Four plus three, well
that's equal to seven. That's equal to an odd number. And so it looks like if I
have an even plus an odd, and I've only tried out two cases here, but I encourage you to try
out many, many, many more to make sure that you
feel good about this, but an even and an odd is
always going to give you an odd. So this is always going
to give you an odd. So let's just remind ourselves, even numbers, they're multiples
of two, including zero. Odd numbers are just numbers
that aren't multiples of two, that aren't even. And an even plus an even is
going to be equal to an even. An odd plus an odd is also
going to be equal to an even. We saw that, we saw that multiple times. But an even plus an odd? That is odd.