Division and remainders

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Notas de aula

What is division?

Division is splitting a number into equal groups. For example, 12 ÷ 3 means: how many groups of 3 are in 12?
The dividend is the number being divided (the big number). The divisor is the number you divide by (the group size). The quotient is the answer.
You can check a division by multiplying: quotient × divisor = dividend. For example, 12 ÷ 3 = 4, and 4 × 3 = 12.

What is a remainder?

Sometimes a number cannot be split into equal groups exactly. The amount left over is called the remainder.
For example, 13 ÷ 5: 5 goes into 13 twice (2 × 5 = 10), and 3 is left over. So 13 ÷ 5 = 2 remainder 3.
The remainder is always less than the divisor. In 13 ÷ 5, the remainder 3 is less than 5.
You can write a division with a remainder like this: 13 ÷ 5 = 2 R 3. The 'R' means remainder.

Division as sharing and grouping

Sharing means splitting a number into equal groups. For example, share 12 sweets among 3 friends: each gets 4.
Grouping means finding how many groups of a certain size you can make. For example, how many groups of 4 are in 12? Answer: 3 groups.
Both sharing and grouping use division, but they look different in real life.
When there is a remainder, it means you cannot share or group exactly. For example, share 13 sweets among 5 friends: each gets 2, and 3 sweets are left over.

Short division (formal method)

Short division is a quick way to divide a two-digit or three-digit number by a one-digit number.
Write the dividend inside the division bracket and the divisor outside, like this: 5 ⟌ 73.
Divide the first digit: 7 ÷ 5 = 1 remainder 2. Write 1 above the 7, and carry the 2 to the next digit.
Now divide 23 (the carried 2 and the next digit 3) by 5: 23 ÷ 5 = 4 remainder 3. Write 4 above the 3, and the remainder 3 is the final remainder.
So 73 ÷ 5 = 14 R 3. Check: 14 × 5 = 70, plus 3 = 73.

Checking division with multiplication

You can always check a division answer by multiplying the quotient by the divisor and adding the remainder.
The rule is: (quotient × divisor) + remainder = dividend.
For example, 73 ÷ 5 = 14 R 3. Check: (14 × 5) + 3 = 70 + 3 = 73. ✔
If the check does not match, you made a mistake. Go back and try again.

Remainders in real life: round up or down?

In real problems, you often need to decide what to do with the remainder.
Sometimes you round up (add 1 to the quotient) because you need a whole extra group. For example, if 13 people need cars and each car holds 5, you need 3 cars (2 full cars + 1 extra).
Sometimes you round down (ignore the remainder) because you only care about full groups. For example, if you have 13 sweets and each bag holds 5, you can fill 2 full bags.
Always read the question carefully to decide: does the remainder mean you need one more, or can you leave it out?

Factor pairs and multiples

A factor pair is two numbers that multiply to give a target number. For example, 3 and 4 are a factor pair of 12 because 3 × 4 = 12.
A multiple of a number is what you get when you multiply it by a whole number. For example, multiples of 5 are 5, 10, 15, 20, ...
If a number is a multiple of the divisor, the remainder is 0. For example, 15 ÷ 5 = 3 R 0, because 15 is a multiple of 5.
Finding factor pairs helps you see if a division will have a remainder. For example, 12 has factor pairs (1,12), (2,6), (3,4), so 12 ÷ 3 = 4 exactly.

Practice tips

Always write down the remainder clearly, like 'R 3'.
Check your answer with multiplication every time.
When dividing, remember: the remainder must be smaller than the divisor. If it is not, your quotient is too small.
Try real-life problems: sharing snacks, packing items, or grouping objects. This helps you understand remainders.

Slides

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Questões de prática

Prévia grátis — 8 de 65 perguntas. Cadastre-se para ver todas.

1.What is the name for the amount left over after dividing one whole number by another?
Easy
- Aremainder
- Bquotient
- Cdivisor
- Dproduct
2.In the division 43 ÷ 5, the quotient is 8 and the remainder is 3. Which equation shows this correctly?
Easy
- A43 = 8 × 5 + 3
- B43 = 8 × 5 − 3
- C43 = 8 + 5 + 3
- D43 = 8 × 3 + 5
3.The remainder is always less than the divisor.
Easy
True or false?
4.Which of the following are true about the division 43 ÷ 5? (Select all that apply)
Medium
- A43 = 8 × 5 + 3
- BThe remainder is 3
- C43 = 9 × 5 − 2
- DThe remainder is 2
5.Match each division to its remainder.
Medium
- 17 ÷ 3
- 20 ÷ 4
- 25 ÷ 6
- 2
- 0
- 1
6.Put these numbers in order from smallest to largest.
Medium
- remainder of 29 ÷ 6
- remainder of 33 ÷ 5
- remainder of 40 ÷ 7
7.A class has 34 students. They are put into groups of 4. How many groups are there, and how many students are left over?
Medium
- A8 groups, 2 left over
- B8 groups, 3 left over
- C9 groups, 0 left over
- D7 groups, 6 left over
8.When dividing 43 by 5, the least absolute remainder is −2.
Medium
True or false?

Mais perguntas

9.Which of the following are true about remainders? (Select all that apply)
Medium
- AThe remainder is always less than the divisor.
- BThe remainder can be equal to the divisor.
- CThe remainder is the amount left over after division.
- DThe remainder can be negative when using least absolute remainder.
10.The least positive remainder is always non-negative.
Easy
True or false?
11.Match each division to its quotient and remainder.
Medium
- 39 ÷ 4
- 50 ÷ 7
- 23 ÷ 3
- 9 R 3
- 7 R 1
- 7 R 2
12.Order the divisions by their remainder from smallest to largest.
Medium
- 17 ÷ 5
- 26 ÷ 4
- 31 ÷ 6
13.Order these divisions by their remainder from smallest to largest.
Medium
- 31 ÷ 5
- 44 ÷ 6
- 50 ÷ 7
- 33 ÷ 4
14.Which of the following divisions have a remainder of 3? (Select all that apply)
Medium
- A47 ÷ 6
- B51 ÷ 8
- C38 ÷ 5
- D29 ÷ 4
- E56 ÷ 9
15.When you divide a number by 7, the remainder can be 7.
Easy
True or false?
16.Put these divisions in order from smallest quotient to largest quotient.
Medium
- 93 ÷ 9
- 74 ÷ 8
- 55 ÷ 6
17.Which of the following statements are true? (Select all that apply)
Medium
- AWhen dividing by 6, the remainder can be 6.
- BThe remainder is always less than the divisor.
- CIf you divide 100 by 7, the remainder is 2.
- DIn 45 ÷ 8, the quotient is 5 and the remainder is 5.
- EIf a number is divisible by 9, the remainder is 0.
18.A box holds 8 pencils. You have 50 pencils. How many boxes are completely filled, and how many pencils are left over?
Medium
- A6 boxes, 2 left over
- B6 boxes, 4 left over
- C7 boxes, 2 left over
- D7 boxes, 4 left over
19.When you divide 53 by 7, the quotient is 7 and the remainder is 4.
Easy
True or false?
20.Which of these divisions have a remainder of 1? (Select all that apply)
Medium
- A22 ÷ 3
- B31 ÷ 5
- C40 ÷ 6
- D49 ÷ 8
- E55 ÷ 9
21.Match each division to its correct quotient and remainder.
Medium
- 47 ÷ 6
- 58 ÷ 7
- 73 ÷ 8
- 7 R5
- 8 R2
- 9 R1
22.Order these divisions from smallest remainder to largest remainder.
Medium
- 34 ÷ 5
- 41 ÷ 6
- 29 ÷ 4
23.Which division equation is correct for 61 divided by 9?
Medium
- A61 = 6 × 9 + 7
- B61 = 7 × 9 − 2
- C61 = 6 × 9 + 5
- D61 = 7 × 9 + 2
24.The remainder is always smaller than the divisor.
Easy
True or false?
25.A teacher has 83 stickers and gives each student 4 stickers. How many students get stickers, and how many stickers are left over?
Medium
- A20 students, 3 left over
- B21 students, 1 left over
- C20 students, 2 left over
- D21 students, 0 left over
26.Which of the following are true statements about division? (Select all that apply)
Medium
- AThe quotient is the result of division.
- BThe remainder is always greater than the divisor.
- CThe remainder can be zero.
- DThe remainder is always a whole number.
- EThe remainder is always less than the divisor.
27.Order these divisions from smallest quotient to largest quotient.
Medium
- 47 ÷ 5
- 58 ÷ 7
- 63 ÷ 8
28.A baker has 74 muffins and puts them in boxes of 6. How many boxes are completely filled, and how many muffins are left over?
Medium
- A12 boxes, 2 left over
- B12 boxes, 4 left over
- C13 boxes, 2 left over
- D13 boxes, 4 left over
29.The least absolute remainder when 43 is divided by 5 is −2.
Medium
True or false?
30.Match each division to its least positive remainder.
Medium
- 29 ÷ 4
- 38 ÷ 5
- 47 ÷ 6
- 1
- 3
- 5
31.A teacher has 73 stickers and wants to give each student 5 stickers. How many students can get stickers, and how many stickers are left over?
Medium
- A14 students, 3 stickers left
- B14 students, 2 stickers left
- C15 students, 3 stickers left
- D15 students, 2 stickers left
32.A baker makes 89 cookies. She puts them in boxes of 6. How many boxes does she need to pack all the cookies?
Medium
- A14 boxes
- B15 boxes
- C13 boxes
- D16 boxes
33.A rectangular garden has an area of 96 square meters. The length is 12 meters. What is the width? Does the division have a remainder?
Hard
- AWidth = 8 m, no remainder
- BWidth = 8 m, remainder 1
- CWidth = 9 m, no remainder
- DWidth = 9 m, remainder 1
34.Which of these divisions have a remainder of 4? (Select all that apply)
Medium
- A58 ÷ 6
- B67 ÷ 9
- C49 ÷ 5
- D73 ÷ 8
- E82 ÷ 7
35.When you divide 85 by 9, the quotient is 9 and the remainder is 4.
Medium
True or false?
36.Put these divisions in order from smallest remainder to largest remainder.
Medium
- 34 ÷ 5
- 29 ÷ 4
- 50 ÷ 6
37.In the division equation 57 ÷ 8 = 7 R 1, what is the number called that is left over?
Easy
- Aquotient
- Bremainder
- Cdivisor
- Ddividend
38.The remainder is the amount left over after dividing one number by another.
Easy
True or false?
39.What is the remainder when 64 is divided by 9?
Easy
- A7
- B1
- C8
- D0
40.In the division 43 = 8 × 5 + 3, the number 8 is the quotient.
Easy
True or false?
41.Divide 58 by 7. What is the remainder?
Medium
42.Complete the sentence with the correct word.
Easy
In the division equation 43 = 8 × 5 + 3, the number 8 is called the ____ and the number 3 is called the ____.
43.What is the remainder when 42 is divided by 5?
Easy
44.What is the quotient?
Easy
45.What is the remainder?
Easy
46.What is the remainder when 100 is divided by 9?
Medium
47.Complete the equation to show the division.
Medium
The division 43 ÷ 5 can be written as 43 = ____ × 5 + ____.
48.What is the remainder when 47 is divided by 6? Write your answer as a number.
Medium
49.A baker has 50 cookies and puts them in boxes of 6. How many boxes are completely filled?
Medium
50.A teacher has 62 stickers and gives 8 stickers to each student. How many students get stickers, and how many stickers are left over? Enter the number of students.
Easy
51.A school bus has 45 seats. If 6 children sit on each bench, how many benches are completely filled and how many children are on the partially filled bench? Write your answer as 'benches filled, children left over'.
Medium
52.What is the remainder when 37 is divided by 5?
Easy
53.Explain why the remainder must always be less than the divisor.
Medium
54.When dividing a number by 8, what are the possible remainders?
Hard
55.Explain why, when you divide a whole number by 5, the remainder cannot be 5 or more.
Hard
56.Explain why the remainder must always be less than the divisor. Use an example in your explanation.
Hard
57.When dividing a number by 7, what are the possible remainders? Explain your answer.
Medium
58.Divide 58 by 7. What is the remainder?
Medium
- A1
- B2
- C3
- D4
59.What is the remainder when 42 is divided by 5?
Easy
- A3
- B2
- C1
- D4
60.What is the quotient?
Easy
- A10
- B9
- C7
- D8
61.What is the remainder?
Easy
- A5
- B2
- C3
- D4
62.What is the remainder when 100 is divided by 9?
Medium
- A2
- B0
- C3
- D1
63.A baker has 50 cookies and puts them in boxes of 6. How many boxes are completely filled?
Medium
- A10 boxes
- B8 boxes
- C7 boxes
- D9 boxes
64.A teacher has 62 stickers and gives 8 stickers to each student. How many students get stickers, and how many stickers are left over? Enter the number of students.
Easy
- A9
- B8
- C7
- D6
65.What is the remainder when 37 is divided by 5?
Easy
- A3
- B2
- C4
- D1

Resumos de revisão

## Division and Remainders – Quick Revision ### What is division? Division is **sharing** or **grouping** a number into equal parts. The number you divide is the **dividend**. The number you divide by is the **divisor**. The answer is the **quotient**. ### What is a remainder? Sometimes division does not work out exactly. The amount **left over** is called the **remainder**. For example, 43 ÷ 5: 5 goes into 43 eight times (8 × 5 = 40), and 3 is left over. So 43 ÷ 5 = 8 remainder 3. We write: 43 = 8 × 5 + 3. **Rule:** The remainder is always **less than the divisor**. It cannot be equal to or bigger than the divisor. ### How to do short division (formal method) 1. Write the dividend inside the division bracket, divisor outside. 2. Divide the first digit (or first two digits if needed) by the divisor. 3. Write the quotient digit above. 4. Multiply the quotient digit by the divisor, write the product under the dividend digits. 5. Subtract to find the remainder. 6. Bring down the next digit and repeat. 7. When no digits are left, the final remainder is the answer's remainder. **Example:** 73 ÷ 3 - 3 into 7 goes 2 (2 × 3 = 6), remainder 1. - Bring down 3 → 13. 3 into 13 goes 4 (4 × 3 = 12), remainder 1. - So 73 ÷ 3 = 24 remainder 1. Check: 24 × 3 + 1 = 72 + 1 = 73. ### Checking division with multiplication To check a division with remainder: **quotient × divisor + remainder = dividend**. If this is true, your answer is correct. ### Real-life problems: round up or down? When a remainder is left, decide what to do with it based on the question: - If you need **whole groups** (e.g., number of buses), round **up**. - If you only care about **full groups** (e.g., number of full boxes), round **down**. **Example:** 50 children, 8 per bus. 50 ÷ 8 = 6 remainder 2. You need 7 buses (round up). ### Factor pairs and multiples - **Factor pairs** of a number are two numbers that multiply to give that number. For 12: (1,12), (2,6), (3,4). - A **multiple** of a number is the result of multiplying it by a whole number. Multiples of 4: 4, 8, 12, 16, … - If a number divides another with no remainder, it is a **factor** of that number. ### Key formula For division with remainder: **Dividend = Quotient × Divisor + Remainder** Always remember: Remainder < Divisor. Check your answer with multiplication.

dividend: The number being divided. In 43 ÷ 5, 43 is the dividend.
divisor: The number you divide by. In 43 ÷ 5, 5 is the divisor.
quotient: The result of division, ignoring any remainder. In 43 ÷ 5, the quotient is 8.
remainder: The amount left over after division. In 43 ÷ 5, the remainder is 3.
factor pair: Two numbers that multiply together to make a given number. For 12, factor pairs are (1,12), (2,6), and (3,4).
multiple: The result of multiplying a number by a whole number. Multiples of 3 include 3, 6, 9, 12, etc.

Flashcards

What is a remainder in division?

The remainder is the amount left over after dividing. For example, 43 ÷ 5 = 8 with remainder 3, because 5 × 8 = 40 and 43 - 40 = 3.

In the division 43 ÷ 5, what is the quotient and what is the remainder?

The quotient is 8 and the remainder is 3. This is because 5 × 8 = 40 and 43 - 40 = 3.

What does the equation a = q × d + r mean in division?

It means: a is the number you divide (dividend), d is the number you divide by (divisor), q is the quotient, and r is the remainder. The remainder r is always less than the divisor d and not negative.

When you divide 37 by 6, what is the quotient and remainder?

37 ÷ 6 = 6 with remainder 1, because 6 × 6 = 36 and 37 - 36 = 1.

How do you check a division with a remainder using multiplication?

Multiply the quotient by the divisor, then add the remainder. You should get the dividend. For example, 8 × 5 + 3 = 43.

What is the rule for the remainder when dividing by a number?

The remainder must be less than the divisor and not negative. For example, when dividing by 7, the remainder can be 0, 1, 2, 3, 4, 5, or 6.

Divide 89 by 4. What is the quotient and remainder?

89 ÷ 4 = 22 with remainder 1, because 4 × 22 = 88 and 89 - 88 = 1.

What does 'division as sharing' mean?

Division as sharing means splitting a number into equal groups. For example, sharing 12 sweets among 3 friends gives each 4 sweets. If there is a remainder, some sweets are left over.

What does 'division as grouping' mean?

Division as grouping means finding how many groups of a certain size you can make. For example, how many groups of 5 are in 17? You get 3 groups and 2 left over.

In a real problem, when do you round a remainder up?

You round up when you need to have enough for everyone or everything. For example, if 23 children need seats and each bus holds 8, you need 3 buses because 2 buses hold 16 and 3 buses hold 24.

In a real problem, when do you round a remainder down?

You round down when you can only use full groups. For example, if you have 23 apples and each bag holds 8, you can fill 2 full bags and have 7 apples left over.

What is short division?

Short division is a quick way to divide by a one-digit number. You write the dividend and divisor, divide digit by digit, and carry remainders to the next digit. For example, 84 ÷ 3: 8 ÷ 3 = 2 remainder 2, then 24 ÷ 3 = 8, so the answer is 28.

Divide 72 by 5 using short division. What is the quotient and remainder?

72 ÷ 5 = 14 with remainder 2. Because 5 × 14 = 70 and 72 - 70 = 2.

What is a factor pair?

A factor pair is two numbers that multiply together to give a product. For example, 3 and 4 are a factor pair of 12 because 3 × 4 = 12.

List all factor pairs of 24.

The factor pairs of 24 are: 1 and 24, 2 and 12, 3 and 8, 4 and 6.

What is a multiple of a number?

A multiple of a number is the result of multiplying that number by a whole number. For example, multiples of 4 are 4, 8, 12, 16, 20, ...

Is 36 a multiple of 9? How do you know?

Yes, 36 is a multiple of 9 because 9 × 4 = 36. So when you divide 36 by 9, the remainder is 0.

What is the remainder when you divide 100 by 7?

100 ÷ 7 = 14 with remainder 2, because 7 × 14 = 98 and 100 - 98 = 2.

What is the least positive remainder?

The least positive remainder is the smallest non-negative remainder you get when dividing. For example, 43 ÷ 5 gives remainder 3, which is the least positive remainder.

What is the least absolute remainder?

The least absolute remainder is the remainder closest to zero, which can be negative. For example, 43 ÷ 5 can also be written as 9 × 5 - 2, so the least absolute remainder is -2.

How do you find the remainder using long division?

In long division, you divide, multiply, subtract, and bring down digits. The final number left after all digits are used is the remainder. For example, 85 ÷ 4: 4 goes into 8 twice (2), multiply 2×4=8, subtract 8-8=0, bring down 5, 4 goes into 5 once (1), multiply 1×4=4, subtract 5-4=1, so remainder is 1.

What does the modulo operation do?

The modulo operation finds the remainder when you divide one number by another. For example, 17 mod 5 = 2, because 17 ÷ 5 = 3 with remainder 2.

If you have 50 pencils and you put them in packs of 6, how many full packs can you make and how many pencils are left?

You can make 8 full packs because 6 × 8 = 48, and 2 pencils are left over. So 50 ÷ 6 = 8 remainder 2.

A teacher has 35 stickers and wants to give each student 4 stickers. How many students can get stickers and how many stickers are left?

8 students can get 4 stickers each because 4 × 8 = 32, and 3 stickers are left. So 35 ÷ 4 = 8 remainder 3.

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