Not every division comes out exactly. When the divisor doesn't fit a whole number of times, there is a remainder — what's left over when no group can be made any larger.
Sharing sweets
You have 23 sweets and want to share them equally among 4 children.
- Each child gets 5 sweets (4 × 5 = 20).
- 3 sweets are left over — that's the remainder.
We write 23 ÷ 4 = 5 remainder 3, or 5 r 3 for short.
The remainder is always smaller than the divisor
If the remainder were equal to or bigger than the divisor, the divisor would fit one more time and the quotient should be one bigger.
23 ÷ 4 = 5 r 3 ✓ (3 < 4)
23 ÷ 4 = 4 r 7 ✗ (7 > 4 — wrong! 7 still contains another 4)
Always check: remainder < divisor.
Checking with the inverse
You can verify a division-with-remainder by going the other way:
quotient × divisor + remainder = dividend
Example: 23 ÷ 4 = 5 r 3. Check:
5 × 4 + 3 = 20 + 3 = 23 ✓
If the result equals the original dividend, you divided correctly. Always do this if you're unsure.
A multi-digit example
752 ÷ 6 = ?
| Step | Current piece | How many 6s? | Quotient digit | Subtraction |
| 1 | 7 | 1 (6 × 1 = 6) | 1 | 7 − 6 = 1 |
| 2 | 15 (bring down 5) | 2 (6 × 2 = 12) | 2 | 15 − 12 = 3 |
| 3 | 32 (bring down 2) | 5 (6 × 5 = 30) | 5 | 32 − 30 = 2 |
Result: 752 ÷ 6 = 125 r 2.
Check: 125 × 6 + 2 = 750 + 2 = 752 ✓