Class 6 Maths worksheet on LCM and HCF with problems to find HCF and LCM of given numbers.

Let's solve each problem step by step.

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Part 1: Find the HCF (Highest Common Factor)

The HCF of two or more numbers is the largest number that divides all of them exactly.

We'll use prime factorization or Euclidean algorithm to find HCF.

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#### 1. HCF of 144 and 198

Prime factorization:

- 144 = 2 × 2 × 2 × 2 × 3 × 3 = $ 2^4 \times 3^2 $
- 198 = 2 × 3 × 3 × 11 = $ 2 \times 3^2 \times 11 $

Common factors: $ 2 \times 3^2 = 2 \times 9 = 18 $

✔ HCF = 18

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#### 2. HCF of 81 and 117

- 81 = 3 × 3 × 3 × 3 = $ 3^4 $
- 117 = 3 × 3 × 13 = $ 3^2 \times 13 $

Common factors: $ 3^2 = 9 $

✔ HCF = 9

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#### 3. HCF of 84 and 98

- 84 = 2 × 2 × 3 × 7 = $ 2^2 \times 3 \times 7 $
- 98 = 2 × 7 × 7 = $ 2 \times 7^2 $

Common factors: $ 2 \times 7 = 14 $

✔ HCF = 14

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#### 4. HCF of 225 and 450

Note: 450 is divisible by 225 → 450 ÷ 225 = 2

So, HCF = 225

Alternatively:
- 225 = $ 3^2 \times 5^2 $
- 450 = $ 2 \times 3^2 \times 5^2 $

Common factors: $ 3^2 \times 5^2 = 9 \times 25 = 225 $

✔ HCF = 225

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#### 5. HCF of 170 and 238

- 170 = 2 × 5 × 17
- 238 = 2 × 7 × 17

Common factors: $ 2 \times 17 = 34 $

✔ HCF = 34

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Part 2: Find the LCM (Least Common Multiple)

LCM is the smallest number divisible by all given numbers.

Use prime factorization and take the highest powers of all primes.

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#### 6. LCM of 48 and 60

- 48 = $ 2^4 \times 3 $
- 60 = $ 2^2 \times 3 \times 5 $

Take highest powers:
- $ 2^4 $, $ 3^1 $, $ 5^1 $

LCM = $ 2^4 \times 3 \times 5 = 16 \times 3 \times 5 = 240 $

✔ LCM = 240

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#### 7. LCM of 42 and 63

- 42 = $ 2 \times 3 \times 7 $
- 63 = $ 3^2 \times 7 $

Highest powers:
- $ 2^1 $, $ 3^2 $, $ 7^1 $

LCM = $ 2 \times 9 \times 7 = 126 $

✔ LCM = 126

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#### 8. LCM of 18 and 17

- 18 = $ 2 \times 3^2 $
- 17 = $ 17 $ (prime)

No common factors → LCM = $ 18 \times 17 = 306 $

✔ LCM = 306

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#### 9. LCM of 15, 30, and 90

Let’s find prime factors:

- 15 = $ 3 \times 5 $
- 30 = $ 2 \times 3 \times 5 $
- 90 = $ 2 \times 3^2 \times 5 $

Highest powers:
- $ 2^1 $, $ 3^2 $, $ 5^1 $

LCM = $ 2 \times 9 \times 5 = 90 $

✔ LCM = 90

(Also, since 90 is a multiple of both 15 and 30, it's the LCM.)

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#### 10. LCM of 56, 65, and 85

First, factorize:

- 56 = $ 2^3 \times 7 $
- 65 = $ 5 \times 13 $
- 85 = $ 5 \times 17 $

Now, take highest powers of all primes:

- $ 2^3 $, $ 5^1 $, $ 7^1 $, $ 13^1 $, $ 17^1 $

LCM = $ 8 \times 5 \times 7 \times 13 \times 17 $

Step-by-step:

- $ 8 \times 5 = 40 $
- $ 40 \times 7 = 280 $
- $ 280 \times 13 = 3640 $
- $ 3640 \times 17 $

Calculate:
$ 3640 \times 17 = 3640 \times (10 + 7) = 36400 + 25480 = 61880 $

✔ LCM = 61,880

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✔ Final Answers:

| Question | Answer |
|--------|--------|
| 1. HCF of 144, 198 | 18 |
| 2. HCF of 81, 117 | 9 |
| 3. HCF of 84, 98 | 14 |
| 4. HCF of 225, 450 | 225 |
| 5. HCF of 170, 238 | 34 |
| 6. LCM of 48, 60 | 240 |
| 7. LCM of 42, 63 | 126 |
| 8. LCM of 18, 17 | 306 |
| 9. LCM of 15, 30, 90 | 90 |
| 10. LCM of 56, 65, 85 | 61,880 |

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