Odd and Even Numbers

Can you and a friend share a bag of sweets with none left over? Can everyone in the class grab a partner for a game? Whether the answer is a tidy "yes" comes down to one thing: whether the number is even or odd.

Every whole number is either even or odd. The test is simple: try to pair the things up two-by-two. If they pair up perfectly with none left over, the number is even. If there is always one left over, the number is odd.

An even number also splits cleanly into two equal groups — that is the same idea seen from the other side. Counting the even numbers is just skip counting by 2:

2,\ 4,\ 6,\ 8,\ 10,\ \dots

And the odd numbers are the ones in between — every other number, starting from one:

1,\ 3,\ 5,\ 7,\ 9,\ \dots

Press play, then replay it: each time it takes a new random number of dots and tries to pair them up. Watch whether the last dot finds a partner, or stands alone.

Even: everyone has a partner

Even numbers love company — every single thing finds a partner. Here are 6 ducks. Line them up two-by-two and they make three tidy pairs with nobody left out, so 6 is even:

duck duck duck duck duck duck = 3 pairs, none left over

Splitting into two equal groups is the very same idea: those 6 ducks share fairly between two ponds — 3 ducks each. An even number can always be shared between two with nothing left behind.

Even numbers are the "fair-sharing" numbers. Give two children 8 cookies and they each get exactly 4 — no squabbling, no last cookie to fight over. That is what "splits into two equal groups" means, and it is only ever possible when the number is even.

cookie cookie cookie cookie | cookie cookie cookie cookie = 4 each

Odd: always one left out

Odd numbers can never pair up completely — there is always a lonely one with no partner. Here are 5 fish. They make two pairs, and then one fish is left over swimming alone, so 5 is odd:

fish fish fish fish fish ← one left over

If you tried to share 5 fish between two tanks you would get 2 each and one fish stuck in your hand. That leftover is the signature of an odd number — it can never be split into two equal groups.

Walk along the counting numbers and they alternate, forever: 1 odd, 2 even, 3 odd, 4 even, 5 odd… Every even number has an odd number on each side of it, like beads on a string. That is why adding 1 always flips a number from one kind to the other.

1, 2, 3, 4, 5, 6, 7 — odd, even, odd, even…

See it: line them up in pairs

Here is the pairing test as a picture you can poke at. The counters stack up in columns of two — each ringed column is a complete pair. If the very last counter has no partner (it glows on its own), the number is odd; if every counter is inside a ring, the number is even. Press Refresh for a new number to test.

The quick shortcut: look at the last digit

You do not have to pair up a hundred socks to know if 100 is even. There is a shortcut you can read in a blink: look only at the last digit. A number is even when it ends in 0,\ 2,\ 4,\ 6 or 8, and odd when it ends in 1,\ 3,\ 5,\ 7 or 9.

Let's try a few:

34 ends in 4 → even.
57 ends in 7 → odd.
290 ends in 0 → even — even though it is a big number, the last digit is all you check.
4\,681 ends in 1 → odd.

Why does this work? Because all the tens, hundreds and thousands are already made of pairs (ten is five pairs, a hundred is fifty pairs…). They always pair up perfectly, so the only thing that can ever leave a leftover is the ones digit. Check the ones, and you know the whole number.

Two traps that catch people out:

Only the last digit matters. 134 is even because it ends in 4 — the 1 and 3 in front make no difference at all.
Zero is even. 0 splits into two equal groups (nothing and nothing) with none left over, so it counts as even — and that is why any number ending in 0, like 20 or 70, is even too.

Khan Academy introduces even and odd numbers here: