Worksheet on LCM - Free Printable

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 or the division method (as shown in problem 1) to find the LCM.

---

1) 10 and 24

Given:
```
2 | 10, 24
2 | 5, 12
3 | 5, 6
| 5, 2
```

Wait — there's a mistake in the provided solution. Let's correct it.

Actually, we should divide both numbers together until they are reduced to 1.

Correct division:

```
2 | 10, 24
| 5, 12
2 | 5, 6
| 5, 3
3 | 5, 3
| 5, 1
5 | 5, 1
| 1, 1
```

So, LCM = 2 × 2 × 3 × 5 = 60

But the given answer says 18, which is wrong.

Let’s verify:
- Multiples of 10: 10, 20, 30, 40, 50, 60, ...
- Multiples of 24: 24, 48, 60, ...

Yes, LCM = 60

✔ So, correct LCM = 60

---

2) 27 and 9

We can write:
- 27 = 3³
- 9 = 3²

LCM = highest power of all primes = 3³ = 27

Alternatively, using division:

```
3 | 27, 9
3 | 9, 3
3 | 3, 1
| 1, 1
```

LCM = 3 × 3 × 3 = 27

✔ LCM = 27

---

3) 84 and 12

First, simplify:
- 84 ÷ 12 = 7 → so 12 divides into 84 exactly

Thus, since 84 is a multiple of 12, the LCM is 84

Check:
- Multiples of 12: 12, 24, 36, 48, 60, 72, 84...
- Multiples of 84: 84, 168...

Smallest common = 84

✔ LCM = 84

---

4) 30, 60, and 15

Let’s do prime factorization:
- 30 = 2 × 3 × 5
- 60 = 2² × 3 × 5
- 15 = 3 × 5

Take highest powers:
- 2², 3¹, 5¹ → LCM = 4 × 3 × 5 = 60

Check:
- 60 is divisible by 30? Yes.
- 60 divisible by 60? Yes.
- 60 divisible by 15? Yes.

✔ LCM = 60

---

5) 45, 70, and 25

Prime factorizations:
- 45 = 3² × 5
- 70 = 2 × 5 × 7
- 25 = 5²

Highest powers:
- 2¹, 3², 5², 7¹

LCM = 2 × 9 × 25 × 7 =
= 2 × 9 = 18
18 × 25 = 450
450 × 7 = 3150

✔ LCM = 3150

---

6) 245 and 512

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

No common factors → LCM = product of both

LCM = 2⁹ × 5 × 7² = 512 × 5 × 49

Calculate:
- 512 × 5 = 2560
- 2560 × 49

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

✔ LCM = 125,440

---

7) 875 and 336

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

- 336:
- 336 ÷ 2 = 168
- 168 ÷ 2 = 84
- 84 ÷ 2 = 42
- 42 ÷ 2 = 21
- 21 ÷ 3 = 7
- 7 ÷ 7 = 1
- So, 336 = 2⁴ × 3 × 7

Now take highest powers:
- 2⁴, 3¹, 5³, 7¹

LCM = 16 × 3 × 125 × 7

Step-by-step:
- 16 × 3 = 48
- 48 × 125 = ?
- 48 × 100 = 4800
- 48 × 25 = 1200
- Total = 4800 + 1200 = 6000
- 6000 × 7 = 42,000

✔ LCM = 42,000

---

8) 1225 and 400

Factorize:
- 1225: ends with 25 → divisible by 25
- 1225 ÷ 25 = 49
- 25 = 5², 49 = 7²
- So, 1225 = 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

Calculate:
- 2⁴ = 16
- 5² = 25
- 7² = 49

Now:
- 16 × 25 = 400
- 400 × 49 = ?

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

> Note: The first problem has an error in the image — it incorrectly shows LCM = 18 for 10 and 24. Correct LCM is 60.

Let me know if you want the steps in a different format!