HCF (Highest Common Factor) math worksheet with exercises on identifying factors and solving word problems.

Problem Analysis and Solution

The worksheet focuses on the concept of the Highest Common Factor (HCF). Let's solve each section step by step.

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## Section A: Circle the numbers in the list that are factors of both numbers in the first column

Task:

Circle the numbers in the list that are factors of both numbers in the first column. The first one has been done for you.

#### Rows:
1. 12 and 15
- Factors of 12: \(1, 2, 3, 4, 6, 12\)
- Factors of 15: \(1, 3, 5, 15\)
- Common factors: \(1, 3\)
- Circled: \(1, 3\)

2. 18 and 24
- Factors of 18: \(1, 2, 3, 6, 9, 18\)
- Factors of 24: \(1, 2, 3, 4, 6, 8, 12, 24\)
- Common factors: \(1, 2, 3, 6\)
- Circled: \(1, 2, 3, 6\)

3. 36 and 42
- Factors of 36: \(1, 2, 3, 4, 6, 9, 12, 18, 36\)
- Factors of 42: \(1, 2, 3, 6, 7, 14, 21, 42\)
- Common factors: \(1, 2, 3, 6\)
- Circled: \(1, 2, 3, 6\)

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## Section B: List all the factors of each number

Task:

List all the factors of each number. The first two have been done for you.

#### Numbers:
1. 16
- Factors: \(1, 2, 4, 8, 16\)

2. 20
- Factors: \(1, 2, 4, 5, 10, 20\)

3. 48
- Factors: \(1, 2, 3, 4, 6, 8, 12, 16, 24, 48\)

4. 100
- Factors: \(1, 2, 4, 5, 10, 20, 25, 50, 100\)

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## Section C: Use the table above to work out the following

Task:

Use the table above to find the HCF of the given sets of numbers.

#### Questions:
1. a) The HCF of 16 and 48
- Factors of 16: \(1, 2, 4, 8, 16\)
- Factors of 48: \(1, 2, 3, 4, 6, 8, 12, 16, 24, 48\)
- Common factors: \(1, 2, 4, 8, 16\)
- HCF: \(16\)

2. b) The HCF of 12, 16, and 48
- Factors of 12: \(1, 2, 3, 4, 6, 12\)
- Factors of 16: \(1, 2, 4, 8, 16\)
- Factors of 48: \(1, 2, 3, 4, 6, 8, 12, 16, 24, 48\)
- Common factors: \(1, 2, 4\)
- HCF: \(4\)

3. c) The HCF of 20 and 100
- Factors of 20: \(1, 2, 4, 5, 10, 20\)
- Factors of 100: \(1, 2, 4, 5, 10, 20, 25, 50, 100\)
- Common factors: \(1, 2, 4, 5, 10, 20\)
- HCF: \(20\)

4. d) The HCF of 50, 75, and 100
- Factors of 50: \(1, 2, 5, 10, 25, 50\)
- Factors of 75: \(1, 3, 5, 15, 25, 75\)
- Factors of 100: \(1, 2, 4, 5, 10, 20, 25, 50, 100\)
- Common factors: \(1, 5, 25\)
- HCF: \(25\)

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## Section D: Word Problems

Problem 1: Ruby's Party Bags

Ruby is preparing party bags for her birthday party. She wants to include 42 marbles and 28 cupcakes. Each party bag must have the same number of marbles and cupcakes as the other bags. She wants to create the greatest number of party bags possible. How many party bags will she use?

#### Solution:
To determine the greatest number of party bags, we need to find the HCF of 42 and 28.

- Factors of 42: \(1, 2, 3, 6, 7, 14, 21, 42\)
- Factors of 28: \(1, 2, 4, 7, 14, 28\)
- Common factors: \(1, 2, 7, 14\)
- HCF: \(14\)

Thus, Ruby can make 14 party bags, with each bag containing:
- \( \frac{42}{14} = 3 \) marbles
- \( \frac{28}{14} = 2 \) cupcakes

Answer: \(\boxed{14}\)

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Problem 2: School Trip Groups

On a school trip, a teacher must put pupils into identical groups based on their school year. 54 Year 4 pupils, 108 Year 5 pupils, and 180 Year 6 pupils go on the trip.

#### Subtasks:
1. a) What is the greatest number of groups she can create?
2. b) How many pupils of each school year will be within each group?

#### Solution:
To determine the greatest number of groups, we need to find the HCF of 54, 108, and 180.

- Factors of 54: \(1, 2, 3, 6, 9, 18, 27, 54\)
- Factors of 108: \(1, 2, 3, 4, 6, 9, 12, 18, 27, 36, 54, 108\)
- Factors of 180: \(1, 2, 3, 4, 5, 6, 9, 10, 12, 15, 18, 20, 30, 36, 45, 60, 90, 180\)
- Common factors: \(1, 2, 3, 6, 9, 18\)
- HCF: \(18\)

Thus, the greatest number of groups is 18.

#### b) Pupils per Group:
- Year 4 pupils per group: \( \frac{54}{18} = 3 \)
- Year 5 pupils per group: \( \frac{108}{18} = 6 \)
- Year 6 pupils per group: \( \frac{180}{18} = 10 \)

Answers:
- a) \(\boxed{18}\)
- b) 3 Year 4 pupils, 6 Year 5 pupils, and 10 Year 6 pupils per group.

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## Final Answers:
1. Section A:
- 12 and 15: \(1, 3\)
- 18 and 24: \(1, 2, 3, 6\)
- 36 and 42: \(1, 2, 3, 6\)

2. Section B:
- 16: \(1, 2, 4, 8, 16\)
- 20: \(1, 2, 4, 5, 10, 20\)
- 48: \(1, 2, 3, 4, 6, 8, 12, 16, 24, 48\)
- 100: \(1, 2, 4, 5, 10, 20, 25, 50, 100\)

3. Section C:
- a) HCF of 16 and 48: \(\boxed{16}\)
- b) HCF of 12, 16, and 48: \(\boxed{4}\)
- c) HCF of 20 and 100: \(\boxed{20}\)
- d) HCF of 50, 75, and 100: \(\boxed{25}\)

4. Section D:
- Ruby's Party Bags: \(\boxed{14}\)
- School Trip Groups:
- a) \(\boxed{18}\)
- b) 3 Year 4 pupils, 6 Year 5 pupils, and 10 Year 6 pupils per group.