Comprehensive math worksheet designed to help students revise key integer concepts like division, factors, and roots.

Let's solve each section of the Integers Revision worksheet step by step and explain the solutions clearly.

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1. Long Division and Multiplication

#### a) $ 6744 \div 4 $
We divide 6744 by 4:

- $ 4 \times 1600 = 6400 $
- $ 6744 - 6400 = 344 $
- $ 4 \times 80 = 320 $
- $ 344 - 320 = 24 $
- $ 4 \times 6 = 24 $

So, $ 1600 + 80 + 6 = 1686 $

✔ Answer: 1686

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#### b) $ 3956 \times 14 $

Break it down:
$$
3956 \times 14 = 3956 \times (10 + 4) = 3956 \times 10 + 3956 \times 4
$$
- $ 3956 \times 10 = 39560 $
- $ 3956 \times 4 = 15824 $
- Add: $ 39560 + 15824 = 55384 $

✔ Answer: 55384

---

#### c) $ 25493 \times 7 $

$$
25493 \times 7 = ?
$$

Calculate:
- $ 7 \times 3 = 21 $ → write 1, carry 2
- $ 7 \times 9 = 63 + 2 = 65 $ → write 5, carry 6
- $ 7 \times 4 = 28 + 6 = 34 $ → write 4, carry 3
- $ 7 \times 5 = 35 + 3 = 38 $ → write 8, carry 3
- $ 7 \times 2 = 14 + 3 = 17 $

So, result is 178451

✔ Answer: 178451

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#### d) $ 5145 \div 7 $

Divide:
- $ 7 \times 700 = 4900 $
- $ 5145 - 4900 = 245 $
- $ 7 \times 35 = 245 $

So, $ 700 + 35 = 735 $

✔ Answer: 735

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#### e) $ 6024 \div 3 $

- $ 3 \times 2000 = 6000 $
- $ 6024 - 6000 = 24 $
- $ 3 \times 8 = 24 $

So, $ 2000 + 8 = 2008 $

✔ Answer: 2008

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#### f) $ 4825 \div 5 $

- $ 5 \times 900 = 4500 $
- $ 4825 - 4500 = 325 $
- $ 5 \times 65 = 325 $

So, $ 900 + 65 = 965 $

✔ Answer: 965

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#### g) $ 7286 \times 4 $

- $ 4 \times 6 = 24 $ → write 4, carry 2
- $ 4 \times 8 = 32 + 2 = 34 $ → write 4, carry 3
- $ 4 \times 2 = 8 + 3 = 11 $ → write 1, carry 1
- $ 4 \times 7 = 28 + 1 = 29 $

Result: 29144

✔ Answer: 29144

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#### h) $ 18592 \times 7 $

- $ 7 \times 2 = 14 $ → write 4, carry 1
- $ 7 \times 9 = 63 + 1 = 64 $ → write 4, carry 6
- $ 7 \times 5 = 35 + 6 = 41 $ → write 1, carry 4
- $ 7 \times 8 = 56 + 4 = 60 $ → write 0, carry 6
- $ 7 \times 1 = 7 + 6 = 13 $

Result: 130144

✔ Answer: 130144

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✔ Summary of Long Division & Multiplication:

| Problem | Answer |
|--------|--------|
| a) $ 6744 \div 4 $ | 1686 |
| b) $ 3956 \times 14 $ | 55384 |
| c) $ 25493 \times 7 $ | 178451 |
| d) $ 5145 \div 7 $ | 735 |
| e) $ 6024 \div 3 $ | 2008 |
| f) $ 4825 \div 5 $ | 965 |
| g) $ 7286 \times 4 $ | 29144 |
| h) $ 18592 \times 7 $ | 130144 |

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## 2. Finding Lowest Common Multiple (LCM)

We find LCM using prime factorization or listing multiples.

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#### a) 8 and 5

- Prime factors:
- $ 8 = 2^3 $
- $ 5 = 5 $
- LCM = $ 2^3 \times 5 = 8 \times 5 = 40 $

✔ LCM: 40

---

#### b) 7 and 3

- Both are primes
- LCM = $ 7 \times 3 = 21 $

✔ LCM: 21

---

#### c) 24 and 3

- $ 24 = 2^3 \times 3 $
- $ 3 = 3 $
- LCM = $ 2^3 \times 3 = 24 $

✔ LCM: 24

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#### d) 15 and 2

- $ 15 = 3 \times 5 $
- $ 2 = 2 $
- LCM = $ 2 \times 3 \times 5 = 30 $

✔ LCM: 30

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#### e) 4, 5, and 7

- All are co-prime
- LCM = $ 4 \times 5 \times 7 = 140 $

✔ LCM: 140

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#### f) 9, 15, and 12

- $ 9 = 3^2 $
- $ 15 = 3 \times 5 $
- $ 12 = 2^2 \times 3 $
- Take highest powers:
- $ 2^2, 3^2, 5 $
- LCM = $ 4 \times 9 \times 5 = 180 $

✔ LCM: 180

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✔ Summary of LCM:

| Problem | Numbers | LCM |
|--------|--------|-----|
| a) | 8, 5 | 40 |
| b) | 7, 3 | 21 |
| c) | 24, 3 | 24 |
| d) | 15, 2 | 30 |
| e) | 4, 5, 7 | 140 |
| f) | 9, 15, 12 | 180 |

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## 3. Finding Highest Common Factor (HCF)

Use prime factorization or Euclidean algorithm.

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#### a) 18 and 30

- $ 18 = 2 \times 3^2 $
- $ 30 = 2 \times 3 \times 5 $
- Common: $ 2 \times 3 = 6 $

✔ HCF: 6

---

#### b) 15 and 40

- $ 15 = 3 \times 5 $
- $ 40 = 2^3 \times 5 $
- Common: $ 5 $

✔ HCF: 5

---

#### c) 21 and 12

- $ 21 = 3 \times 7 $
- $ 12 = 2^2 \times 3 $
- Common: $ 3 $

✔ HCF: 3

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#### d) 27 and 90

- $ 27 = 3^3 $
- $ 90 = 2 \times 3^2 \times 5 $
- Common: $ 3^2 = 9 $

✔ HCF: 9

---

#### e) 39 and 16

- $ 39 = 3 \times 13 $
- $ 16 = 2^4 $
- No common factors → HCF = 1

✔ HCF: 1

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#### f) 100 and 250

- $ 100 = 2^2 \times 5^2 $
- $ 250 = 2 \times 5^3 $
- Common: $ 2 \times 5^2 = 2 \times 25 = 50 $

✔ HCF: 50

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✔ Summary of HCF:

| Problem | Numbers | HCF |
|--------|--------|-----|
| a) | 18, 30 | 6 |
| b) | 15, 40 | 5 |
| c) | 21, 12 | 3 |
| d) | 27, 90 | 9 |
| e) | 39, 16 | 1 |
| f) | 100, 250 | 50 |

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## 4. Squares and Square Roots

Now compute:

#### a) $ 5^2 = 25 $

#### b) $ \sqrt{4} = 2 $

#### c) $ 2^2 = 4 $

#### d) $ 7^2 = 49 $

#### e) $ \sqrt{9} = 3 $

#### f) $ 4^2 = 16 $

#### g) $ \sqrt{25} = 5 $

#### h) $ \sqrt{36} = 6 $

#### i) $ \sqrt[3]{8} = 2 $ (since $ 2^3 = 8 $)

#### j) $ 8^2 = 64 $

#### k) $ \sqrt[3]{125} = 5 $ (since $ 5^3 = 125 $)

#### l) $ 9^2 = 81 $

#### m) $ 11^2 = 121 $

#### n) $ \sqrt[3]{1000} = 10 $ (since $ 10^3 = 1000 $)

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✔ Final Answers for Squares and Roots:

| Problem | Answer |
|--------|--------|
| a) $ 5^2 $ | 25 |
| b) $ \sqrt{4} $ | 2 |
| c) $ 2^2 $ | 4 |
| d) $ 7^2 $ | 49 |
| e) $ \sqrt{9} $ | 3 |
| f) $ 4^2 $ | 16 |
| g) $ \sqrt{25} $ | 5 |
| h) $ \sqrt{36} $ | 6 |
| i) $ \sqrt[3]{8} $ | 2 |
| j) $ 8^2 $ | 64 |
| k) $ \sqrt[3]{125} $ | 5 |
| l) $ 9^2 $ | 81 |
| m) $ 11^2 $ | 121 |
| n) $ \sqrt[3]{1000} $ | 10 |

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## ✔ Final Answer Sheet

Long Division and Multiplication

a) 1686
b) 55384
c) 178451
d) 735
e) 2008
f) 965
g) 29144
h) 130144

LCM

a) 40
b) 21
c) 24
d) 30
e) 140
f) 180

HCF

a) 6
b) 5
c) 3
d) 9
e) 1
f) 50

Squares and Roots

a) 25
b) 2
c) 4
d) 49
e) 3
f) 16
g) 5
h) 6
i) 2
j) 64
k) 5
l) 81
m) 121
n) 10

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