Least Common Multiples Worksheets - 15 Worksheets Library - Free Printable

To solve the problem of finding the Least Common Multiple (LCM) for each pair of numbers, we will use the prime factorization method. Here's a step-by-step explanation for each pair:

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1. (20, 30)

#### Step 1: Prime Factorization
- 20:
$$
20 = 2 \times 2 \times 5 = 2^2 \times 5
$$
- 30:
$$
30 = 2 \times 3 \times 5
$$

#### Step 2: Identify the Highest Powers of All Prime Factors
- The prime factors involved are: $2$, $3$, and $5$.
- For $2$: The highest power is $2^2$ (from 20).
- For $3$: The highest power is $3^1$ (from 30).
- For $5$: The highest power is $5^1$ (common in both).

#### Step 3: Multiply the Highest Powers
$$
\text{LCM} = 2^2 \times 3^1 \times 5^1 = 4 \times 3 \times 5 = 60
$$

#### Final Answer:
$$
\boxed{60}
$$

---

2. (10, 24)

#### Step 1: Prime Factorization
- 10:
$$
10 = 2 \times 5
$$
- 24:
$$
24 = 2 \times 2 \times 2 \times 3 = 2^3 \times 3
$$

#### Step 2: Identify the Highest Powers of All Prime Factors
- The prime factors involved are: $2$, $3$, and $5$.
- For $2$: The highest power is $2^3$ (from 24).
- For $3$: The highest power is $3^1$ (from 24).
- For $5$: The highest power is $5^1$ (from 10).

#### Step 3: Multiply the Highest Powers
$$
\text{LCM} = 2^3 \times 3^1 \times 5^1 = 8 \times 3 \times 5 = 120
$$

#### Final Answer:
$$
\boxed{120}
$$

---

3. (12, 30)

#### Step 1: Prime Factorization
- 12:
$$
12 = 2 \times 2 \times 3 = 2^2 \times 3
$$
- 30:
$$
30 = 2 \times 3 \times 5
$$

#### Step 2: Identify the Highest Powers of All Prime Factors
- The prime factors involved are: $2$, $3$, and $5$.
- For $2$: The highest power is $2^2$ (from 12).
- For $3$: The highest power is $3^1$ (common in both).
- For $5$: The highest power is $5^1$ (from 30).

#### Step 3: Multiply the Highest Powers
$$
\text{LCM} = 2^2 \times 3^1 \times 5^1 = 4 \times 3 \times 5 = 60
$$

#### Final Answer:
$$
\boxed{60}
$$

---

4. (15, 25)

#### Step 1: Prime Factorization
- 15:
$$
15 = 3 \times 5
$$
- 25:
$$
25 = 5 \times 5 = 5^2
$$

#### Step 2: Identify the Highest Powers of All Prime Factors
- The prime factors involved are: $3$ and $5$.
- For $3$: The highest power is $3^1$ (from 15).
- For $5$: The highest power is $5^2$ (from 25).

#### Step 3: Multiply the Highest Powers
$$
\text{LCM} = 3^1 \times 5^2 = 3 \times 25 = 75
$$

#### Final Answer:
$$
\boxed{75}
$$

---

5. (30, 10)

#### Step 1: Prime Factorization
- 30:
$$
30 = 2 \times 3 \times 5
$$
- 10:
$$
10 = 2 \times 5
$$

#### Step 2: Identify the Highest Powers of All Prime Factors
- The prime factors involved are: $2$, $3$, and $5$.
- For $2$: The highest power is $2^1$ (common in both).
- For $3$: The highest power is $3^1$ (from 30).
- For $5$: The highest power is $5^1$ (common in both).

#### Step 3: Multiply the Highest Powers
$$
\text{LCM} = 2^1 \times 3^1 \times 5^1 = 2 \times 3 \times 5 = 30
$$

#### Final Answer:
$$
\boxed{30}
$$

---

6. (25, 45)

#### Step 1: Prime Factorization
- 25:
$$
25 = 5 \times 5 = 5^2
$$
- 45:
$$
45 = 3 \times 3 \times 5 = 3^2 \times 5
$$

#### Step 2: Identify the Highest Powers of All Prime Factors
- The prime factors involved are: $3$ and $5$.
- For $3$: The highest power is $3^2$ (from 45).
- For $5$: The highest power is $5^2$ (from 25).

#### Step 3: Multiply the Highest Powers
$$
\text{LCM} = 3^2 \times 5^2 = 9 \times 25 = 225
$$

#### Final Answer:
$$
\boxed{225}
$$

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Final Answers for All Pairs

1. $(20, 30)$: $\boxed{60}$
2. $(10, 24)$: $\boxed{120}$
3. $(12, 30)$: $\boxed{60}$
4. $(15, 25)$: $\boxed{75}$
5. $(30, 10)$: $\boxed{30}$
6. $(25, 45)$: $\boxed{225}$

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$$
\boxed{60, 120, 60, 75, 30, 225}
$$