Division with Remainders
Learn division with remainders: what a remainder is, how to find it, why the remainder is always smaller than the divisor, and how to write answers — with worked examples and a quiz.
Key takeaways
- A remainder is what is left over when a number does not divide exactly
- The remainder is always smaller than the number you are dividing by
- Write it as a whole number answer plus 'r' and the remainder, e.g. 17 ÷ 5 = 3 r 2
- You can check division by multiplying back and adding the remainder
When sharing does not come out even
Sometimes a division works out perfectly: 12 ÷ 3 = 4, with nothing left over. But often it does not. If you try to share 13 biscuits between 4 friends, each friend gets 3 biscuits and there is 1 biscuit left over. That leftover is called a remainder.
If you are confident with Division Made Simple, this lesson adds the missing piece: what to do when the numbers do not split evenly.
What is a remainder?
A remainder is the amount left over after you have shared into equal groups as far as you can. We write it with the letter r:
13 ÷ 4 = 3 r 1
This reads: "13 divided by 4 is 3, remainder 1." It means each group gets 3, and 1 is left over.
How to find a remainder
Use the times table closest to your number without going over.
Example — 17 ÷ 5:
- Think: how many 5s fit into 17? 5 × 3 = 15, but 5 × 4 = 20, which is too big.
- So 5 goes in 3 times, using up 15.
- What is left? 17 − 15 = 2.
- So 17 ÷ 5 = 3 r 2.
The golden rule
The remainder is always smaller than the number you are dividing by.
Why? If you are dividing by 5 and the leftover reached 5, you could make one more whole group — so it would not really be left over. You keep grouping until what remains is too small to form another group.
| Division | Groups of | Goes in | Used | Remainder |
|---|---|---|---|---|
| 14 ÷ 3 | 3 | 4 | 12 | 2 |
| 25 ÷ 4 | 4 | 6 | 24 | 1 |
| 30 ÷ 7 | 7 | 4 | 28 | 2 |
| 19 ÷ 6 | 6 | 3 | 18 | 1 |
| 22 ÷ 5 | 5 | 4 | 20 | 2 |
In every row, check that the remainder is less than the "groups of" number. It always is.
Checking your answer
You can always check division with remainders by working backwards:
divisor × answer + remainder = the number you started with
For 17 ÷ 5 = 3 r 2: 5 × 3 + 2 = 15 + 2 = 17. ✔ It matches, so the answer is correct.
Worked example
A teacher has 29 pencils to share equally between 6 pots. How many in each pot, and how many are left over?
- How many 6s in 29? 6 × 4 = 24, and 6 × 5 = 30 is too big. So 6 goes in 4 times.
- Used: 24 pencils. Left over: 29 − 24 = 5.
- So 29 ÷ 6 = 4 r 5 — each pot gets 4 pencils, with 5 left over.
- Check: 6 × 4 + 5 = 24 + 5 = 29. ✔
Try it yourself
- Work out: 11 ÷ 2, 23 ÷ 4, 31 ÷ 5. Write each answer as "_ r _".
- Check one of your answers using divisor × answer + remainder.
- Real life: 25 children get into teams of 4. How many full teams, and how many children are left without a full team?
(Answers: 11 ÷ 2 = 5 r 1; 23 ÷ 4 = 5 r 3; 31 ÷ 5 = 6 r 1. Teams: 6 full teams, 1 child left over.)
Where this leads
Remainders are the stepping stone to Long Division, where you carry leftovers across to the next digit. They also connect to Factors and Multiples: a number divides another exactly only when the remainder is zero. Keep practising, and remainders will feel natural.
Quick quiz
Test yourself and earn XP
What is 13 ÷ 4?
4 goes into 13 three times (4 × 3 = 12), leaving 13 − 12 = 1. So 13 ÷ 4 = 3 r 1.
When you divide by 5, what is the largest remainder you could ever get?
The remainder must be smaller than the divisor. Dividing by 5, the biggest remainder is 4 — if it reached 5, that would be another whole group.
What is 20 ÷ 6?
6 × 3 = 18, and 20 − 18 = 2, so 20 ÷ 6 = 3 r 2.
23 ÷ 7 = 3 r 2. How can you check this is right?
Multiply the answer by the divisor and add the remainder: 7 × 3 + 2 = 21 + 2 = 23. It matches!
Which division has NO remainder?
18 ÷ 3 = 6 exactly, because 18 is a multiple of 3. The other two leave something over.
FAQ
A remainder is the amount left over when one number does not divide exactly into another. For example, 17 sweets shared between 5 children gives 3 each with 2 left over — the remainder is 2.
If the remainder were as big as the divisor, you could make one more whole group. So you keep sharing until what is left is too small to form another group, which means the remainder is always less than the divisor.
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