Math worksheet focusing on H.C.F. and L.C.M. problems, including exercises for finding the highest common factor and lowest common multiple using various methods.

Let's solve each problem step-by-step from the worksheet on H.C.F. (Highest Common Factor) and L.C.M. (Lowest Common Multiple).

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I. Find H.C.F. by Complete Factorization

We'll factor each number into its prime factors and then identify the common ones with the lowest powers.

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#### (i) 48, 56, 72

Prime factorization:
- 48 = $ 2^4 \times 3 $
- 56 = $ 2^3 \times 7 $
- 72 = $ 2^3 \times 3^2 $

Common factors: Only $ 2^3 $ is common to all three.

✔ H.C.F. = $ 2^3 = 8 $
But the answer given in the worksheet is (2) — this seems incorrect.

Wait! Let's check again:
- 48: $ 2^4 \times 3 $
- 56: $ 2^3 \times 7 $
- 72: $ 2^3 \times 3^2 $

Common prime factor: only 2, and minimum power is $ 2^3 $. So H.C.F. = 8

✘ But the worksheet says (2) — that must be a typo or error.

👉 Correct H.C.F. = 8

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#### (ii) 198, 360

Factorize:
- 198 = $ 2 \times 3^2 \times 11 $
- 360 = $ 2^3 \times 3^2 \times 5 $

Common factors: $ 2^1 \times 3^2 = 2 \times 9 = 18 $

✔ H.C.F. = 18

Worksheet says (2) — again, incorrect.

👉 Correct H.C.F. = 18

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#### (iii) 102, 68, 136

- 102 = $ 2 \times 3 \times 17 $
- 68 = $ 2^2 \times 17 $
- 136 = $ 2^3 \times 17 $

Common factors: $ 2^1 \times 17 = 34 $

✔ H.C.F. = 34

Worksheet says (2) — incorrect.

👉 Correct H.C.F. = 34

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#### (iv) 1024, 576

- 1024 = $ 2^{10} $
- 576 = $ 2^6 \times 3^2 $

Common: $ 2^6 = 64 $

✔ H.C.F. = 64

Worksheet says (2) — wrong.

👉 Correct H.C.F. = 64

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#### (v) 405, 783, 513

Factorize:
- 405 = $ 3^4 \times 5 $
- 783 = $ 3^2 \times 29 $
- 513 = $ 3^3 \times 19 $

Common factor: $ 3^2 = 9 $

✔ H.C.F. = 9

Worksheet says (2) — incorrect.

👉 Correct H.C.F. = 9

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> 🔴 Note: All answers marked as "(2)" in the worksheet seem to be incorrect. They likely indicate the number of marks, not the actual answer. The numbers in red parentheses are probably marks allocated, not the answers.

So we proceed assuming those are marks, not answers.

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II. Find H.C.F. by Long Division Method

We use repeated division method.

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#### (i) 84, 144

Step 1: Divide 144 by 84
- 144 ÷ 84 = 1 remainder 60
- 84 ÷ 60 = 1 rem 24
- 60 ÷ 24 = 2 rem 12
- 24 ÷ 12 = 2 rem 0

✔ H.C.F. = 12

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#### (ii) 120, 168

- 168 ÷ 120 = 1 rem 48
- 120 ÷ 48 = 2 rem 24
- 48 ÷ 24 = 2 rem 0

✔ H.C.F. = 24

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#### (iii) 430, 516, 817

We find H.C.F. of two at a time.

First, H.C.F. of 430 and 516:

- 516 ÷ 430 = 1 rem 86
- 430 ÷ 86 = 5 rem 0 → H.C.F. = 86

Now H.C.F. of 86 and 817:

- 817 ÷ 86 = 9 rem 43
- 86 ÷ 43 = 2 rem 0 → H.C.F. = 43

✔ H.C.F. = 43

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#### (iv) 632, 790, 869

Start with 632 and 790:

- 790 ÷ 632 = 1 rem 158
- 632 ÷ 158 = 4 rem 0 → H.C.F. = 158

Now H.C.F. of 158 and 869:

- 869 ÷ 158 = 5 rem 79
- 158 ÷ 79 = 2 rem 0 → H.C.F. = 79

✔ H.C.F. = 79

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#### (v) 291, 582, 776

First: 582 ÷ 291 = 2 rem 0 → H.C.F. = 291

Now H.C.F. of 291 and 776:

- 776 ÷ 291 = 2 rem 194
- 291 ÷ 194 = 1 rem 97
- 194 ÷ 97 = 2 rem 0 → H.C.F. = 97

✔ H.C.F. = 97

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#### (vi) 219, 1321, 2320, 8526

Do step-by-step:

Start with 219 and 1321:

- 1321 ÷ 219 = 6 rem 7 (since 219×6=1314)
- 219 ÷ 7 = 31 rem 2
- 7 ÷ 2 = 3 rem 1
- 2 ÷ 1 = 2 rem 0 → H.C.F. = 1

So H.C.F. of 219 and 1321 is 1

Now check if any other number shares a common factor with 1 → no.

✔ H.C.F. = 1

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III. Find L.C.M. of the following numbers

We use prime factorization method.

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#### (i) 16, 24, 40

Factorize:
- 16 = $ 2^4 $
- 24 = $ 2^3 \times 3 $
- 40 = $ 2^3 \times 5 $

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

L.C.M. = $ 2^4 \times 3 \times 5 = 16 \times 3 \times 5 = 240 $

✔ L.C.M. = 240

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#### (ii) 40, 56, 60

- 40 = $ 2^3 \times 5 $
- 56 = $ 2^3 \times 7 $
- 60 = $ 2^2 \times 3 \times 5 $

Max powers:
- $ 2^3 $, $ 3^1 $, $ 5^1 $, $ 7^1 $

L.C.M. = $ 8 \times 3 \times 5 \times 7 = 840 $

✔ L.C.M. = 840

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#### (iii) 207, 138

- 207 = $ 3^2 \times 23 $
- 138 = $ 2 \times 3 \times 23 $

Take max powers:
- $ 2^1 $, $ 3^2 $, $ 23^1 $

L.C.M. = $ 2 \times 9 \times 23 = 18 \times 23 = 414 $

✔ L.C.M. = 414

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## ✔ Final Answers Summary

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I. H.C.F. by Factorization

| Problem | Numbers | H.C.F. |
|--------|--------|--------|
| (i) | 48, 56, 72 | 8 |
| (ii) | 198, 360 | 18 |
| (iii) | 102, 68, 136 | 34 |
| (iv) | 1024, 576 | 64 |
| (v) | 405, 783, 513 | 9 |

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II. H.C.F. by Long Division

| Problem | Numbers | H.C.F. |
|--------|--------|--------|
| (i) | 84, 144 | 12 |
| (ii) | 120, 168 | 24 |
| (iii) | 430, 516, 817 | 43 |
| (iv) | 632, 790, 869 | 79 |
| (v) | 291, 582, 776 | 97 |
| (vi) | 219, 1321, 2320, 8526 | 1 |

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III. L.C.M.

| Problem | Numbers | L.C.M. |
|--------|--------|--------|
| (i) | 16, 24, 40 | 240 |
| (ii) | 40, 56, 60 | 840 |
| (iii) | 207, 138 | 414 |

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> 🔍 Important Note: The numbers in red parentheses like (2) or (3) are likely marks allocated, not answers. Do not confuse them with actual solutions.

Let me know if you'd like these problems solved using a different method or explained further!