Division Basics • Topic 3 of 3

Remainders

Sometimes a number cannot be shared into equal groups exactly — a little is left over. The amount left over is called the remainder.

Example. Share $13$ sweets among $4$ children. Each child gets $3$ (that uses $12$ sweets), and $1$ sweet is left over. We write this as:

$$13\div4 = 3 \text{ remainder } 1\quad(\text{short: } 3\text{ r }1)$$

Here $3$ is the quotient (how many each got) and $1$ is the remainder (what was left).

The most important rule: the remainder is always smaller than the divisor. If the remainder were as big as the divisor, you could give out one more to each group.

How to check your answer. Use:

$$\text{quotient}\times\text{divisor}+\text{remainder}=\text{dividend}$$

For $13\div4=3\text{ r }1$: $3\times4+1=13$ — correct!

When the remainder is $0$. If nothing is left over, the number divides exactly. For example $12\div4=3$ with remainder $0$.

Common mistakes to avoid:

  • Leaving a remainder that is equal to or bigger than the divisor (it must be smaller).
  • Forgetting the remainder when the sharing is not exact.
REMAINDERS:  13 sweets shared among 4 children

   [o o o] [o o o] [o o o] [o o o]   and  o  left over

   Each child gets 3, and 1 is left.
   13 / 4 = 3 remainder 1   (written  3 r 1)

RULE:  the remainder is always SMALLER than the divisor.

CHECK:  quotient x divisor + remainder = dividend
        3 x 4 + 1 = 13  [correct]
Solved Examples
1
Worked Example

Find $13\div4$.

Solution
  • $4\times3=12$, and $13-12=1$ is left.
  • So $13\div4=3\text{ r }1$.
  • Answer: $3$ remainder $1$
2
Worked Example

Find $17\div5$.

Solution
  • $5\times3=15$, and $17-15=2$ is left.
  • So $17\div5=3\text{ r }2$.
  • Answer: $3$ remainder $2$
3
Worked Example

Find $20\div6$.

Solution
  • $6\times3=18$, and $20-18=2$ is left.
  • So $20\div6=3\text{ r }2$.
  • Answer: $3$ remainder $2$
4
Worked Example

Find $25\div4$.

Solution
  • $4\times6=24$, and $25-24=1$ is left.
  • So $25\div4=6\text{ r }1$.
  • Answer: $6$ remainder $1$
5
Worked Example

$14$ sweets are shared among $4$ children. How many does each get, and how many are left over?

Solution
  • $14\div4=3\text{ r }2$.
  • Each child gets $3$; $2$ are left over.
  • Answer: $3$ each, $2$ left
6
Worked Example

Check that $22\div5=4\text{ r }2$ is correct.

Solution
  • Use quotient $\times$ divisor $+$ remainder.
  • $4\times5+2=20+2=22$.
  • It matches the dividend.
  • Answer: Yes, correct

Key Points

  • A remainder is the amount left over when sharing is not exact.
  • Write it as "quotient r remainder", e.g. $13\div4=3\text{ r }1$.
  • The remainder is always smaller than the divisor.
  • Check with: quotient $\times$ divisor $+$ remainder $=$ dividend.
  • If nothing is left over, the remainder is $0$ and the number divides exactly.
  • In word problems, the remainder is whatever cannot be shared equally.
Quick Check — 4 MCQs
Tap an option to check your answer0 / 4
Q1.The amount left over after equal sharing is the:
Explanation: Remainder.
Q2.$13\div4=$
Explanation: $4\times3=12$, $1$ left.
Q3.A remainder is always ___ the divisor.
Explanation: Smaller.
Q4.To check a division with a remainder, you compute:
Explanation: $q\times d+r$.
Practice more MCQs → 📝 Assignment · 35 marks