Remainders

Imagine you have 7 cookies to share fairly between 3 friends. You deal them out one at a time, round and round: one each, one each — and each friend ends up with 2 cookies. But look! There is still 1 cookie sitting on the plate. It can't be shared without breaking it, so it is left over.

cookie cookie cookie cookie cookie cookie + cookie = 1 left over

That left-over part is called the remainder. When division doesn't come out even, the remainder tells you how much is left behind.

Writing it down

Take 13 \div 4. Deal 13 counters into 4 groups, round and round. Each group fills up with 3 counters, which uses up 12 — but 1 is left over with nowhere to go. So:

13 \div 4 = 3 \text{ remainder } 1

We read that as “three remainder one”: three in each group, and one left over. Grown-ups often shorten “remainder” to a little r and write 13 \div 4 = 3 \text{ r } 1. They mean exactly the same thing.

Press play to deal a pile of dots out one at a time, round and round the groups. The groups fill up evenly — and the dots that can’t finish another full round are the remainder, glowing on their own.

See it: deal and find the leftovers

Here is a fresh pile of counters dealt into equal groups. The groups each hold the same amount, and any counters that couldn't make another full group sit below on their own, glowing — that's the remainder. Press Refresh for a brand-new sharing each time.

Three worked examples

Once you can deal into groups and spot the leftovers, every one works the same way:

17 \div 5: five groups, 3 in each uses 15, and 2 are left over — 17 \div 5 = 3 \text{ r } 2.
20 \div 3: three groups, 6 in each uses 18, and 2 are left over — 20 \div 3 = 6 \text{ r } 2.
12 \div 4: four groups, 3 in each uses all 12 — nothing left over, so 12 \div 4 = 3 \text{ r } 0. We usually just write 12 \div 4 = 3.

Two rules that catch everybody out:

The remainder must be less than the number you are dividing by. If your “left over” pile is as big as a group — or bigger — you forgot to deal out another full round. With 4 groups the remainder can only be 0, 1, 2 or 3, never 4.
A remainder of 0 means it shared out exactly — no leftovers at all. That's ordinary division with nothing to write after the “r”.

Deal 8 strawberries into 3 bowls, one at a time. Each bowl gets 2 (that's 6 used up), and 2 strawberries are left on the table — not enough to give every bowl a third one. So 8 \div 3 = 2 \text{ r } 2.

strawberry + = 2 left over

Yes! Share 6 bananas between 2 monkeys and each gets exactly 3 — the plate is empty, nothing is left. When the sharing comes out perfectly even, the remainder is 0, and we don't bother writing it. A remainder of zero is the divider's way of saying “it fits perfectly”.

banana banana banana monkey · banana banana banana monkey = 0 left over

Khan Academy introduces remainders here: