Worksheet on L.C.M. | Least Common Multiple Worksheets |LCM Worksheets - Free Printable

Worksheet on LCM

Let's solve each problem step by step.

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Problem 1: Find the first 3 common multiples of the given using a number line.

#### (i) 3 and 4
- Multiples of 3: $ 3, 6, 9, 12, 15, 18, 21, 24, \ldots $
- Multiples of 4: $ 4, 8, 12, 16, 20, 24, 28, \ldots $

The common multiples are the numbers that appear in both lists:
- The first common multiple is $ 12 $.
- The second common multiple is $ 24 $.
- The third common multiple is $ 36 $.

Thus, the first 3 common multiples of 3 and 4 are:
$$
\boxed{12, 24, 36}
$$

#### (ii) 5 and 6
- Multiples of 5: $ 5, 10, 15, 20, 25, 30, 35, 40, 45, 50, 55, 60, \ldots $
- Multiples of 6: $ 6, 12, 18, 24, 30, 36, 42, 48, 54, 60, \ldots $

The common multiples are:
- The first common multiple is $ 30 $.
- The second common multiple is $ 60 $.
- The third common multiple is $ 90 $.

Thus, the first 3 common multiples of 5 and 6 are:
$$
\boxed{30, 60, 90}
$$

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Problem 2: Find the LCM of the given numbers by prime factorization method.

#### (i) 12 and 20
1. Prime factorization:
- $ 12 = 2^2 \times 3 $
- $ 20 = 2^2 \times 5 $

2. Identify the highest powers of all prime factors:
- For $ 2 $: The highest power is $ 2^2 $.
- For $ 3 $: The highest power is $ 3^1 $.
- For $ 5 $: The highest power is $ 5^1 $.

3. LCM:
$$
\text{LCM} = 2^2 \times 3^1 \times 5^1 = 4 \times 3 \times 5 = 60
$$

Thus, the LCM of 12 and 20 is:
$$
\boxed{60}
$$

#### (ii) 36, 27, and 18
1. Prime factorization:
- $ 36 = 2^2 \times 3^2 $
- $ 27 = 3^3 $
- $ 18 = 2^1 \times 3^2 $

2. Identify the highest powers of all prime factors:
- For $ 2 $: The highest power is $ 2^2 $.
- For $ 3 $: The highest power is $ 3^3 $.

3. LCM:
$$
\text{LCM} = 2^2 \times 3^3 = 4 \times 27 = 108
$$

Thus, the LCM of 36, 27, and 18 is:
$$
\boxed{108}
$$

#### (iii) 32 and 48
1. Prime factorization:
- $ 32 = 2^5 $
- $ 48 = 2^4 \times 3^1 $

2. Identify the highest powers of all prime factors:
- For $ 2 $: The highest power is $ 2^5 $.
- For $ 3 $: The highest power is $ 3^1 $.

3. LCM:
$$
\text{LCM} = 2^5 \times 3^1 = 32 \times 3 = 96
$$

Thus, the LCM of 32 and 48 is:
$$
\boxed{96}
$$

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Problem 3: Find the L.C.M. by Long Division Method.

#### (i) 70, 110, 150
1. Write the numbers in a row:
$$
70, 110, 150
$$

2. Divide by the smallest prime number (2):
$$
70 \div 2 = 35, \quad 110 \div 2 = 55, \quad 150 \div 2 = 75
$$
$$
\text{New row: } 35, 55, 75
$$

3. Divide by the next smallest prime number (5):
$$
35 \div 5 = 7, \quad 55 \div 5 = 11, \quad 75 \div 5 = 15
$$
$$
\text{New row: } 7, 11, 15
$$

4. Divide by the next smallest prime number (3):
- 7 is not divisible by 3.
- 11 is not divisible by 3.
- $ 15 \div 3 = 5 $
$$
\text{New row: } 7, 11, 5
$$

5. Divide by the next smallest prime number (5):
- 7 is not divisible by 5.
- 11 is not divisible by 5.
- $ 5 \div 5 = 1 $
$$
\text{New row: } 7, 11, 1
$$

6. All numbers are now 1, so stop dividing.

7. Calculate the LCM:
$$
\text{LCM} = 2 \times 5 \times 3 \times 7 \times 11 \times 5 = 11550
$$

Thus, the LCM of 70, 110, and 150 is:
$$
\boxed{11550}
$$

#### (ii) 25, 30, 150
1. Write the numbers in a row:
$$
25, 30, 150
$$

2. Divide by the smallest prime number (2):
- 25 is not divisible by 2.
- $ 30 \div 2 = 15 $
- $ 150 \div 2 = 75 $
$$
\text{New row: } 25, 15, 75
$$

3. Divide by the next smallest prime number (3):
- 25 is not divisible by 3.
- $ 15 \div 3 = 5 $
- $ 75 \div 3 = 25 $
$$
\text{New row: } 25, 5, 25
$$

4. Divide by the next smallest prime number (5):
- $ 25 \div 5 = 5 $
- $ 5 \div 5 = 1 $
- $ 25 \div 5 = 5 $
$$
\text{New row: } 5, 1, 5
$$

5. Divide by the next smallest prime number (5):
- $ 5 \div 5 = 1 $
- 1 is already 1.
- $ 5 \div 5 = 1 $
$$
\text{New row: } 1, 1, 1
$$

6. All numbers are now 1, so stop dividing.

7. Calculate the LCM:
$$
\text{LCM} = 2 \times 3 \times 5 \times 5 = 150
$$

Thus, the LCM of 25, 30, and 150 is:
$$
\boxed{150}
$$

#### (iii) 36, 60, 120
1. Write the numbers in a row:
$$
36, 60, 120
$$

2. Divide by the smallest prime number (2):
$$
36 \div 2 = 18, \quad 60 \div 2 = 30, \quad 120 \div 2 = 60
$$
$$
\text{New row: } 18, 30, 60
$$

3. Divide by the next smallest prime number (2):
$$
18 \div 2 = 9, \quad 30 \div 2 = 15, \quad 60 \div 2 = 30
$$
$$
\text{New row: } 9, 15, 30
$$

4. Divide by the next smallest prime number (3):
$$
9 \div 3 = 3, \quad 15 \div 3 = 5, \quad 30 \div 3 = 10
$$
$$
\text{New row: } 3, 5, 10
$$

5. Divide by the next smallest prime number (3):
- $ 3 \div 3 = 1 $
- 5 is not divisible by 3.
- 10 is not divisible by 3.
$$
\text{New row: } 1, 5, 10
$$

6. Divide by the next smallest prime number (5):
- 1 is not divisible by 5.
- $ 5 \div 5 = 1 $
- $ 10 \div 5 = 2 $
$$
\text{New row: } 1, 1, 2
$$

7. Divide by the next smallest prime number (2):
- 1 is not divisible by 2.
- 1 is not divisible by 2.
- $ 2 \div 2 = 1 $
$$
\text{New row: } 1, 1, 1
$$

8. All numbers are now 1, so stop dividing.

9. Calculate the LCM:
$$
\text{LCM} = 2 \times 2 \times 3 \times 3 \times 5 \times 2 = 360
$$

Thus, the LCM of 36, 60, and 120 is:
$$
\boxed{360}
$$

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Final Answers:

1. (i) $ \boxed{12, 24, 36} $
(ii) $ \boxed{30, 60, 90} $

2. (i) $ \boxed{60} $
(ii) $ \boxed{108} $
(iii) $ \boxed{96} $

3. (i) $ \boxed{11550} $
(ii) $ \boxed{150} $
(iii) $ \boxed{360} $