Worksheet for practicing finding the Least Common Multiple (LCM) of given numbers.

Let's solve each of the LCM (Lowest Common Multiple) problems step by step. The LCM of two or more numbers is the smallest positive number that is divisible by all of them.

We'll use the prime factorization method for most cases, and where appropriate, we can also use the division method (as shown in problem 1).

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1) 10 and 24

Given solution:
```
2 | 12, 24
2 | 6, 12
3 | 3, 6
1, 2
```

Wait — this seems to be a mistake! The numbers are 10 and 24, but the division starts with 12 and 24. Let's fix that.

#### Correct approach: Prime factorization

- 10 = 2 × 5
- 24 = 2³ × 3

Take the highest powers of all primes:
- 2³ × 3¹ × 5¹ = 8 × 3 × 5 = 120

✔ LCM = 120

But wait — the image says LCM = 18, which is incorrect. That’s likely a typo.

Let’s double-check:

Multiples of 10: 10, 20, 30, 40, 50, 60, 70, 80, 90, 100, 110, 120, ...

Multiples of 24: 24, 48, 72, 96, 120, ...

So yes, LCM = 120

✘ So the given answer (18) is wrong.

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2) 27 and 9

- 27 = 3³
- 9 = 3²

LCM = highest power of 3 → 3³ = 27

✔ LCM = 27

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3) 84 and 12

- 84 = 2² × 3 × 7
- 12 = 2² × 3

LCM = 2² × 3 × 7 = 4 × 3 × 7 = 84

✔ LCM = 84

(Since 84 is a multiple of 12, it's the LCM.)

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4) 30, 60, and 15

- 30 = 2 × 3 × 5
- 60 = 2² × 3 × 5
- 15 = 3 × 5

Take highest powers:
- 2² × 3 × 5 = 4 × 3 × 5 = 60

✔ LCM = 60

(60 is divisible by all three: 30, 60, 15)

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5) 45, 70, and 25

Prime factorizations:

- 45 = 3² × 5
- 70 = 2 × 5 × 7
- 25 = 5²

Now take highest powers:
- 2¹ × 3² × 5² × 7¹
= 2 × 9 × 25 × 7
= 2 × 9 = 18
18 × 25 = 450
450 × 7 = 3150

✔ LCM = 3150

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6) 245 and 512

- 245 = 5 × 49 = 5 × 7²
- 512 = 2⁹ (since 2^9 = 512)

No common factors → coprime in terms of prime bases

So LCM = 2⁹ × 5 × 7²
= 512 × 5 × 49

Step-by-step:
- 512 × 5 = 2560
- 2560 × 49 = ?

Calculate:
- 2560 × 50 = 128,000
- Subtract 2560 → 128,000 - 2560 = 125,440

✔ LCM = 125,440

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7) 875 and 336

Factor both:

- 875: ends in 5 → divisible by 5
875 ÷ 5 = 175
175 ÷ 5 = 35
35 ÷ 5 = 7
So: 875 = 5³ × 7

- 336: even → divide by 2
336 ÷ 2 = 168
168 ÷ 2 = 84
84 ÷ 2 = 42
42 ÷ 2 = 21
21 ÷ 3 = 7
So: 336 = 2⁴ × 3 × 7

Now LCM = take highest powers:
- 2⁴ × 3 × 5³ × 7

Calculate:
- 2⁴ = 16
- 5³ = 125
- So: 16 × 3 = 48
48 × 125 = ?
48 × 100 = 4800
48 × 25 = 1200 → total = 4800 + 1200 = 6000
Then 6000 × 7 = 42,000

✔ LCM = 42,000

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8) 1225 and 400

Factor:

- 1225: ends in 25 → divisible by 25
1225 ÷ 25 = 49
So: 1225 = 25 × 49 = 5² × 7²

- 400:
400 = 4 × 100 = 2² × 10² = 2² × (2×5)² = 2² × 2² × 5² = 2⁴ × 5²

So:
- 1225 = 5² × 7²
- 400 = 2⁴ × 5²

LCM = 2⁴ × 5² × 7²
= 16 × 25 × 49

Step-by-step:
- 16 × 25 = 400
- 400 × 49 = ?

Calculate:
- 400 × 50 = 20,000
- Subtract 400 → 20,000 - 400 = 19,600

✔ LCM = 19,600

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✔ Final Answers:

| Problem | Numbers | LCM |
|--------|-----------------------|------------|
| 1) | 10 and 24 | 120 |
| 2) | 27 and 9 | 27 |
| 3) | 84 and 12 | 84 |
| 4) | 30, 60, and 15 | 60 |
| 5) | 45, 70, and 25 | 3150 |
| 6) | 245 and 512 | 125,440|
| 7) | 875 and 336 | 42,000 |
| 8) | 1225 and 400 | 19,600 |

> ⚠️ Note: The first problem in the image has an error — it shows LCM = 18, but correct LCM of 10 and 24 is 120.

Let me know if you'd like a visual explanation or step-by-step diagrams for any!