Advanced-Using Prime Factorization to find HCF & LCM worksheet ... - Free Printable

Problem Overview:

The task involves solving problems related to Highest Common Factor (HCF) and Least Common Multiple (LCM) using two methods:
1. Long Division Method for finding HCF.
2. Prime Factorization Method for finding HCF and LCM.

Let's solve each part step by step.

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Part I: Find the HCF/GCF using the Long Division Method

#### 1. 48, 75
- Step 1: Divide the larger number (75) by the smaller number (48).
$$
75 \div 48 = 1 \text{ remainder } 27
$$
So, $ 75 = 48 \times 1 + 27 $.

- Step 2: Now divide the divisor (48) by the remainder (27).
$$
48 \div 27 = 1 \text{ remainder } 21
$$
So, $ 48 = 27 \times 1 + 21 $.

- Step 3: Divide the new divisor (27) by the new remainder (21).
$$
27 \div 21 = 1 \text{ remainder } 6
$$
So, $ 27 = 21 \times 1 + 6 $.

- Step 4: Divide the new divisor (21) by the new remainder (6).
$$
21 \div 6 = 3 \text{ remainder } 3
$$
So, $ 21 = 6 \times 3 + 3 $.

- Step 5: Divide the new divisor (6) by the new remainder (3).
$$
6 \div 3 = 2 \text{ remainder } 0
$$
So, $ 6 = 3 \times 2 + 0 $.

- The remainder is now 0, so the HCF is the last non-zero remainder, which is 3.

Answer: $\boxed{3}$

#### 2. 78, 52, 23
- Step 1: First, find the HCF of 78 and 52 using the long division method.
- Divide 78 by 52:
$$
78 \div 52 = 1 \text{ remainder } 26
$$
So, $ 78 = 52 \times 1 + 26 $.
- Divide 52 by 26:
$$
52 \div 26 = 2 \text{ remainder } 0
$$
So, $ 52 = 26 \times 2 + 0 $.
- The HCF of 78 and 52 is 26.

- Step 2: Now, find the HCF of 26 and 23.
- Divide 26 by 23:
$$
26 \div 23 = 1 \text{ remainder } 3
$$
So, $ 26 = 23 \times 1 + 3 $.
- Divide 23 by 3:
$$
23 \div 3 = 7 \text{ remainder } 2
$$
So, $ 23 = 3 \times 7 + 2 $.
- Divide 3 by 2:
$$
3 \div 2 = 1 \text{ remainder } 1
$$
So, $ 3 = 2 \times 1 + 1 $.
- Divide 2 by 1:
$$
2 \div 1 = 2 \text{ remainder } 0
$$
So, $ 2 = 1 \times 2 + 0 $.
- The HCF of 26 and 23 is 1.

Answer: $\boxed{1}$

#### 3. 84, 96, 63
- Step 1: First, find the HCF of 84 and 96.
- Divide 96 by 84:
$$
96 \div 84 = 1 \text{ remainder } 12
$$
So, $ 96 = 84 \times 1 + 12 $.
- Divide 84 by 12:
$$
84 \div 12 = 7 \text{ remainder } 0
$$
So, $ 84 = 12 \times 7 + 0 $.
- The HCF of 84 and 96 is 12.

- Step 2: Now, find the HCF of 12 and 63.
- Divide 63 by 12:
$$
63 \div 12 = 5 \text{ remainder } 3
$$
So, $ 63 = 12 \times 5 + 3 $.
- Divide 12 by 3:
$$
12 \div 3 = 4 \text{ remainder } 0
$$
So, $ 12 = 3 \times 4 + 0 $.
- The HCF of 12 and 63 is 3.

Answer: $\boxed{3}$

#### 4. 154, 87
- Step 1: Divide 154 by 87:
$$
154 \div 87 = 1 \text{ remainder } 67
$$
So, $ 154 = 87 \times 1 + 67 $.
- Step 2: Divide 87 by 67:
$$
87 \div 67 = 1 \text{ remainder } 20
$$
So, $ 87 = 67 \times 1 + 20 $.
- Step 3: Divide 67 by 20:
$$
67 \div 20 = 3 \text{ remainder } 7
$$
So, $ 67 = 20 \times 3 + 7 $.
- Step 4: Divide 20 by 7:
$$
20 \div 7 = 2 \text{ remainder } 6
$$
So, $ 20 = 7 \times 2 + 6 $.
- Step 5: Divide 7 by 6:
$$
7 \div 6 = 1 \text{ remainder } 1
$$
So, $ 7 = 6 \times 1 + 1 $.
- Step 6: Divide 6 by 1:
$$
6 \div 1 = 6 \text{ remainder } 0
$$
So, $ 6 = 1 \times 6 + 0 $.
- The HCF is 1.

Answer: $\boxed{1}$

#### 5. 96, 54, 75
- Step 1: First, find the HCF of 96 and 54.
- Divide 96 by 54:
$$
96 \div 54 = 1 \text{ remainder } 42
$$
So, $ 96 = 54 \times 1 + 42 $.
- Divide 54 by 42:
$$
54 \div 42 = 1 \text{ remainder } 12
$$
So, $ 54 = 42 \times 1 + 12 $.
- Divide 42 by 12:
$$
42 \div 12 = 3 \text{ remainder } 6
$$
So, $ 42 = 12 \times 3 + 6 $.
- Divide 12 by 6:
$$
12 \div 6 = 2 \text{ remainder } 0
$$
So, $ 12 = 6 \times 2 + 0 $.
- The HCF of 96 and 54 is 6.

- Step 2: Now, find the HCF of 6 and 75.
- Divide 75 by 6:
$$
75 \div 6 = 12 \text{ remainder } 3
$$
So, $ 75 = 6 \times 12 + 3 $.
- Divide 6 by 3:
$$
6 \div 3 = 2 \text{ remainder } 0
$$
So, $ 6 = 3 \times 2 + 0 $.
- The HCF of 6 and 75 is 3.

Answer: $\boxed{3}$

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Part II: Find the HCF/GCF using the Factorisation Method

#### 1. 8, 65, 64
- Step 1: Prime factorize each number.
- $ 8 = 2^3 $
- $ 65 = 5 \times 13 $
- $ 64 = 2^6 $
- Step 2: Identify the common prime factors.
- There are no common prime factors among 8, 65, and 64.
- The HCF is 1.

Answer: $\boxed{1}$

#### 2. 80, 45, 90
- Step 1: Prime factorize each number.
- $ 80 = 2^4 \times 5 $
- $ 45 = 3^2 \times 5 $
- $ 90 = 2 \times 3^2 \times 5 $
- Step 2: Identify the common prime factors.
- The only common prime factor is $ 5 $.
- The HCF is 5.

Answer: $\boxed{5}$

#### 3. 12, 60, 52
- Step 1: Prime factorize each number.
- $ 12 = 2^2 \times 3 $
- $ 60 = 2^2 \times 3 \times 5 $
- $ 52 = 2^2 \times 13 $
- Step 2: Identify the common prime factors.
- The common prime factors are $ 2^2 $.
- The HCF is $ 2^2 = 4 $.

Answer: $\boxed{4}$

#### 4. 11, 121, 110
- Step 1: Prime factorize each number.
- $ 11 = 11 $
- $ 121 = 11^2 $
- $ 110 = 2 \times 5 \times 11 $
- Step 2: Identify the common prime factors.
- The common prime factor is $ 11 $.
- The HCF is 11.

Answer: $\boxed{11}$

#### 5. 24, 48, 32
- Step 1: Prime factorize each number.
- $ 24 = 2^3 \times 3 $
- $ 48 = 2^4 \times 3 $
- $ 32 = 2^5 $
- Step 2: Identify the common prime factors.
- The common prime factor is $ 2^3 $.
- The HCF is $ 2^3 = 8 $.

Answer: $\boxed{8}$

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Part III: Find the LCM using the Factorisation Method

#### 1. 12, 32, 40
- Step 1: Prime factorize each number.
- $ 12 = 2^2 \times 3 $
- $ 32 = 2^5 $
- $ 40 = 2^3 \times 5 $
- Step 2: Take the highest power of each prime factor.
- For $ 2 $: The highest power is $ 2^5 $.
- For $ 3 $: The highest power is $ 3^1 $.
- For $ 5 $: The highest power is $ 5^1 $.
- The LCM is $ 2^5 \times 3^1 \times 5^1 = 32 \times 3 \times 5 = 480 $.

Answer: $\boxed{480}$

#### 2. 40, 54, 62
- Step 1: Prime factorize each number.
- $ 40 = 2^3 \times 5 $
- $ 54 = 2 \times 3^3 $
- $ 62 = 2 \times 31 $
- Step 2: Take the highest power of each prime factor.
- For $ 2 $: The highest power is $ 2^3 $.
- For $ 3 $: The highest power is $ 3^3 $.
- For $ 5 $: The highest power is $ 5^1 $.
- For $ 31 $: The highest power is $ 31^1 $.
- The LCM is $ 2^3 \times 3^3 \times 5^1 \times 31^1 = 8 \times 27 \times 5 \times 31 = 33480 $.

Answer: $\boxed{33480}$

#### 3. 12, 84, 91
- Step 1: Prime factorize each number.
- $ 12 = 2^2 \times 3 $
- $ 84 = 2^2 \times 3 \times 7 $
- $ 91 = 7 \times 13 $
- Step 2: Take the highest power of each prime factor.
- For $ 2 $: The highest power is $ 2^2 $.
- For $ 3 $: The highest power is $ 3^1 $.
- For $ 7 $: The highest power is $ 7^1 $.
- For $ 13 $: The highest power is $ 13^1 $.
- The LCM is $ 2^2 \times 3^1 \times 7^1 \times 13^1 = 4 \times 3 \times 7 \times 13 = 1092 $.

Answer: $\boxed{1092}$

#### 4. 91, 49, 7
- Step 1: Prime factorize each number.
- $ 91 = 7 \times 13 $
- $ 49 = 7^2 $
- $ 7 = 7^1 $
- Step 2: Take the highest power of each prime factor.
- For $ 7 $: The highest power is $ 7^2 $.
- For $ 13 $: The highest power is $ 13^1 $.
- The LCM is $ 7^2 \times 13^1 = 49 \times 13 = 637 $.

Answer: $\boxed{637}$

#### 5. 45, 90, 15
- Step 1: Prime factorize each number.
- $ 45 = 3^2 \times 5 $
- $ 90 = 2 \times 3^2 \times 5 $
- $ 15 = 3 \times 5 $
- Step 2: Take the highest power of each prime factor.
- For $ 2 $: The highest power is $ 2^1 $.
- For $ 3 $: The highest power is $ 3^2 $.
- For $ 5 $: The highest power is $ 5^1 $.
- The LCM is $ 2^1 \times 3^2 \times 5^1 = 2 \times 9 \times 5 = 90 $.

Answer: $\boxed{90}$

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Final Answers:

1. Part I:
- 1. $\boxed{3}$
- 2. $\boxed{1}$
- 3. $\boxed{3}$
- 4. $\boxed{1}$
- 5. $\boxed{3}$

2. Part II:
- 1. $\boxed{1}$
- 2. $\boxed{5}$
- 3. $\boxed{4}$
- 4. $\boxed{11}$
- 5. $\boxed{8}$

3. Part III:
- 1. $\boxed{480}$
- 2. $\boxed{33480}$
- 3. $\boxed{1092}$
- 4. $\boxed{637}$
- 5. $\boxed{90}$

$\boxed{\text{All answers are provided above.}}$