Educational math worksheet on using prime decomposition to find HCF and LCM, featuring matching exercises and Venn diagrams.

Problem Overview:

The task involves using prime decomposition to find the Highest Common Factor (HCF) and Least Common Multiple (LCM) of given numbers. The worksheet is divided into two sections:

1. Section A: Match each number to its product of prime factors.
2. Section B: Use the prime factorizations from Section A to complete Venn diagrams and determine the HCF and LCM for specific pairs of numbers.

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Step 1: Solve Section A

We need to match each number to its correct prime factorization. Let's break it down:

#### Prime Factorization of Each Number:
1. 28:
- \( 28 = 2 \times 14 = 2 \times 2 \times 7 = 2^2 \times 7 \)
- Matches: \( 2^2 \times 7 \)

2. 42:
- \( 42 = 2 \times 21 = 2 \times 3 \times 7 \)
- Matches: \( 2 \times 3 \times 7 \)

3. 56:
- \( 56 = 2 \times 28 = 2 \times 2 \times 14 = 2 \times 2 \times 2 \times 7 = 2^3 \times 7 \)
- Matches: \( 2^3 \times 7 \)

4. 72:
- \( 72 = 2 \times 36 = 2 \times 2 \times 18 = 2 \times 2 \times 2 \times 9 = 2 \times 2 \times 2 \times 3 \times 3 = 2^3 \times 3^2 \)
- Matches: \( 2^3 \times 3^2 \)

5. 80:
- \( 80 = 2 \times 40 = 2 \times 2 \times 20 = 2 \times 2 \times 2 \times 10 = 2 \times 2 \times 2 \times 2 \times 5 = 2^4 \times 5 \)
- Matches: \( 2^4 \times 5 \)

6. 90:
- \( 90 = 2 \times 45 = 2 \times 3 \times 15 = 2 \times 3 \times 3 \times 5 = 2 \times 3^2 \times 5 \)
- Matches: \( 2 \times 3^2 \times 5 \)

7. 98:
- \( 98 = 2 \times 49 = 2 \times 7 \times 7 = 2 \times 7^2 \)
- Matches: \( 2 \times 7^2 \)

8. 112:
- \( 112 = 2 \times 56 = 2 \times 2 \times 28 = 2 \times 2 \times 2 \times 14 = 2 \times 2 \times 2 \times 2 \times 7 = 2^4 \times 7 \)
- Matches: \( 2^4 \times 7 \)

9. 120:
- \( 120 = 2 \times 60 = 2 \times 2 \times 30 = 2 \times 2 \times 2 \times 15 = 2 \times 2 \times 2 \times 3 \times 5 = 2^3 \times 3 \times 5 \)
- Matches: \( 2^3 \times 3 \times 5 \)

#### Final Matches:
- \( 28 \): \( 2^2 \times 7 \)
- \( 42 \): \( 2 \times 3 \times 7 \)
- \( 56 \): \( 2^3 \times 7 \)
- \( 72 \): \( 2^3 \times 3^2 \)
- \( 80 \): \( 2^4 \times 5 \)
- \( 90 \): \( 2 \times 3^2 \times 5 \)
- \( 98 \): \( 2 \times 7^2 \)
- \( 112 \): \( 2^4 \times 7 \)
- \( 120 \): \( 2^3 \times 3 \times 5 \)

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Step 2: Solve Section B

We will use the prime factorizations from Section A to complete the Venn diagrams and find the HCF and LCM for the given pairs.

#### 1) HCF and LCM of 28 and 90

##### Prime Factorizations:
- \( 28 = 2^2 \times 7 \)
- \( 90 = 2 \times 3^2 \times 5 \)

##### Venn Diagram:
- Intersection (Common Factors): \( 2 \)
- Only in 28: \( 2 \times 7 \)
- Only in 90: \( 3^2 \times 5 \)

##### HCF:
The HCF is the product of the common prime factors with the lowest powers.
- Common factor: \( 2 \)
- HCF: \( 2 \)

##### LCM:
The LCM is the product of all prime factors, taking the highest power of each.
- Factors: \( 2^2 \times 3^2 \times 5 \times 7 \)
- LCM: \( 2^2 \times 3^2 \times 5 \times 7 = 4 \times 9 \times 5 \times 7 = 1260 \)

##### Answers:
- HCF of 28 and 90: \( 2 \)
- LCM of 28 and 90: \( 1260 \)

#### 2) HCF and LCM of 28 and 98

##### Prime Factorizations:
- \( 28 = 2^2 \times 7 \)
- \( 98 = 2 \times 7^2 \)

##### Venn Diagram:
- Intersection (Common Factors): \( 2 \times 7 \)
- Only in 28: \( 2 \)
- Only in 98: \( 7 \)

##### HCF:
The HCF is the product of the common prime factors with the lowest powers.
- Common factors: \( 2 \times 7 \)
- HCF: \( 2 \times 7 = 14 \)

##### LCM:
The LCM is the product of all prime factors, taking the highest power of each.
- Factors: \( 2^2 \times 7^2 \)
- LCM: \( 2^2 \times 7^2 = 4 \times 49 = 196 \)

##### Answers:
- HCF of 28 and 98: \( 14 \)
- LCM of 28 and 98: \( 196 \)

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Final Answers:

1. HCF of 28 and 90: \( \boxed{2} \)
LCM of 28 and 90: \( \boxed{1260} \)

2. HCF of 28 and 98: \( \boxed{14} \)
LCM of 28 and 98: \( \boxed{196} \)