Least Common Multiples Worksheets - 15 Worksheets Library - Free Printable

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

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1. Finding LCM of 50 and 75

#### Step 1: Prime Factorization
- 50:
$$
50 \div 2 = 25 \\
25 \div 5 = 5 \\
5 \div 5 = 1
$$
So, the prime factorization of 50 is:
$$
50 = 2 \times 5^2
$$

- 75:
$$
75 \div 3 = 25 \\
25 \div 5 = 5 \\
5 \div 5 = 1
$$
So, the prime factorization of 75 is:
$$
75 = 3 \times 5^2
$$

#### Step 2: Identify the Highest Powers of All Prime Factors
- From 50: $2^1$, $5^2$
- From 75: $3^1$, $5^2$

The highest powers of all prime factors are:
- $2^1$
- $3^1$
- $5^2$

#### Step 3: Calculate the LCM
Multiply these highest powers together:
$$
\text{LCM} = 2^1 \times 3^1 \times 5^2 = 2 \times 3 \times 25 = 150
$$

#### Final Answer:
$$
\boxed{150}
$$

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2. Finding LCM of 40 and 60

#### Step 1: Prime Factorization
- 40:
$$
40 \div 2 = 20 \\
20 \div 2 = 10 \\
10 \div 2 = 5 \\
5 \div 5 = 1
$$
So, the prime factorization of 40 is:
$$
40 = 2^3 \times 5
$$

- 60:
$$
60 \div 2 = 30 \\
30 \div 2 = 15 \\
15 \div 3 = 5 \\
5 \div 5 = 1
$$
So, the prime factorization of 60 is:
$$
60 = 2^2 \times 3 \times 5
$$

#### Step 2: Identify the Highest Powers of All Prime Factors
- From 40: $2^3$, $5^1$
- From 60: $2^2$, $3^1$, $5^1$

The highest powers of all prime factors are:
- $2^3$
- $3^1$
- $5^1$

#### Step 3: Calculate the LCM
Multiply these highest powers together:
$$
\text{LCM} = 2^3 \times 3^1 \times 5^1 = 8 \times 3 \times 5 = 120
$$

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

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3. Finding LCM of 24 and 36

#### Step 1: Prime Factorization
- 24:
$$
24 \div 2 = 12 \\
12 \div 2 = 6 \\
6 \div 2 = 3 \\
3 \div 3 = 1
$$
So, the prime factorization of 24 is:
$$
24 = 2^3 \times 3
$$

- 36:
$$
36 \div 2 = 18 \\
18 \div 2 = 9 \\
9 \div 3 = 3 \\
3 \div 3 = 1
$$
So, the prime factorization of 36 is:
$$
36 = 2^2 \times 3^2
$$

#### Step 2: Identify the Highest Powers of All Prime Factors
- From 24: $2^3$, $3^1$
- From 36: $2^2$, $3^2$

The highest powers of all prime factors are:
- $2^3$
- $3^2$

#### Step 3: Calculate the LCM
Multiply these highest powers together:
$$
\text{LCM} = 2^3 \times 3^2 = 8 \times 9 = 72
$$

#### Final Answer:
$$
\boxed{72}
$$

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4. Finding LCM of 24 and 56

#### Step 1: Prime Factorization
- 24:
$$
24 \div 2 = 12 \\
12 \div 2 = 6 \\
6 \div 2 = 3 \\
3 \div 3 = 1
$$
So, the prime factorization of 24 is:
$$
24 = 2^3 \times 3
$$

- 56:
$$
56 \div 2 = 28 \\
28 \div 2 = 14 \\
14 \div 2 = 7 \\
7 \div 7 = 1
$$
So, the prime factorization of 56 is:
$$
56 = 2^3 \times 7
$$

#### Step 2: Identify the Highest Powers of All Prime Factors
- From 24: $2^3$, $3^1$
- From 56: $2^3$, $7^1$

The highest powers of all prime factors are:
- $2^3$
- $3^1$
- $7^1$

#### Step 3: Calculate the LCM
Multiply these highest powers together:
$$
\text{LCM} = 2^3 \times 3^1 \times 7^1 = 8 \times 3 \times 7 = 168
$$

#### Final Answer:
$$
\boxed{168}
$$

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Summary of Answers

1. LCM of 50 and 75: $\boxed{150}$
2. LCM of 40 and 60: $\boxed{120}$
3. LCM of 24 and 36: $\boxed{72}$
4. LCM of 24 and 56: $\boxed{168}$

These are the final answers. Each step involves prime factorization, identifying the highest powers of all prime factors, and multiplying them to get the LCM.