Factors and multiples simple worksheet | KS3 maths | HCF, LCM ... - Free Printable

Problem Analysis and Solution

The provided worksheet focuses on the concepts of factors and their properties. Let's solve each section step by step.

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#### Section 1: Complete the sentences

1. a. A factor ___________ ___________ into another number. There will be no ___________.
- Solution: A factor divides evenly into another number. There will be no remainder.
- Completed sentence: A factor divides evenly into another number. There will be no remainder.

2. b. The smallest factor of a number is always ___________.
- Solution: The smallest factor of any number is always 1.
- Completed sentence: The smallest factor of a number is always 1.

3. c. The largest factor of a number is always ___________.
- Solution: The largest factor of any number is always the number itself.
- Completed sentence: The largest factor of a number is always the number itself.

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#### Section 2: Write all the factors for each number

To find the factors of a number, list all the numbers that divide it evenly (i.e., with no remainder).

1. a. 4: Factors are \(1, 2, 4\).
2. b. 10: Factors are \(1, 2, 5, 10\).
3. c. 7: Factors are \(1, 7\) (since 7 is prime).
4. d. 12: Factors are \(1, 2, 3, 4, 6, 12\).
5. e. 15: Factors are \(1, 3, 5, 15\).
6. f. 8: Factors are \(1, 2, 4, 8\).
7. g. 2: Factors are \(1, 2\) (since 2 is prime).
8. h. 5: Factors are \(1, 5\) (since 5 is prime).
9. i. 20: Factors are \(1, 2, 4, 5, 10, 20\).
10. j. 9: Factors are \(1, 3, 9\).
11. k. 35: Factors are \(1, 5, 7, 35\).
12. l. 25: Factors are \(1, 5, 25\).
13. m. 24: Factors are \(1, 2, 3, 4, 6, 8, 12, 24\).
14. n. 100: Factors are \(1, 2, 4, 5, 10, 20, 25, 50, 100\).
15. o. 75: Factors are \(1, 3, 5, 15, 25, 75\).
16. p. 81: Factors are \(1, 3, 9, 27, 81\).

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#### Section 3: Complete the missing factors for each number, keeping them in order

1. a. 6: \(1, 2, \_, \_\)
- Factors of 6: \(1, 2, 3, 6\).
- Missing factors: \(3, 6\).
- Completed: \(1, 2, 3, 6\).

2. b. 3: \(\_, \_\)
- Factors of 3: \(1, 3\).
- Missing factors: \(1, 3\).
- Completed: \(1, 3\).

3. c. 14: \(1, \_, 7, \_\)
- Factors of 14: \(1, 2, 7, 14\).
- Missing factors: \(2, 14\).
- Completed: \(1, 2, 7, 14\).

4. d. 16: \(\_, 2, \_, \_, 16\)
- Factors of 16: \(1, 2, 4, 8, 16\).
- Missing factors: \(1, 4, 8\).
- Completed: \(1, 2, 4, 8, 16\).

5. e. 28: \(\_, 2, \_, \_, 28\)
- Factors of 28: \(1, 2, 4, 7, 14, 28\).
- Missing factors: \(1, 4, 7, 14\).
- Completed: \(1, 2, 4, 7, 14, 28\).

6. f. 18: \(\_, \_, 3, \_, 9, \_\)
- Factors of 18: \(1, 2, 3, 6, 9, 18\).
- Missing factors: \(1, 2, 6, 18\).
- Completed: \(1, 2, 3, 6, 9, 18\).

7. g. 30: \(\_, 2, 3, \_, 6, \_, \_, 30\)
- Factors of 30: \(1, 2, 3, 5, 6, 10, 15, 30\).
- Missing factors: \(1, 5, 10, 15\).
- Completed: \(1, 2, 3, 5, 6, 10, 15, 30\).

8. h. 49: \(\_, \_, \_\)
- Factors of 49: \(1, 7, 49\).
- Missing factors: \(1, 7, 49\).
- Completed: \(1, 7, 49\).

9. i. 36: \(\_, 2, 3, \_, \_, 9, \_, \_\)
- Factors of 36: \(1, 2, 3, 4, 6, 9, 12, 18, 36\).
- Missing factors: \(1, 4, 6, 12, 18, 36\).
- Completed: \(1, 2, 3, 4, 6, 9, 12, 18, 36\).

10. j. 90: \(\_, \_, \_, \_, 6, \_, \_, \_, \_, \_\)
- Factors of 90: \(1, 2, 3, 5, 6, 9, 10, 15, 18, 30, 45, 90\).
- Missing factors: \(1, 2, 3, 5, 9, 10, 15, 18, 30, 45, 90\).
- Completed: \(1, 2, 3, 5, 6, 9, 10, 15, 18, 30, 45, 90\).

11. k. 40: \(\_, \_, 4, \_, \_, \_, \_\)
- Factors of 40: \(1, 2, 4, 5, 8, 10, 20, 40\).
- Missing factors: \(1, 2, 5, 8, 10, 20, 40\).
- Completed: \(1, 2, 4, 5, 8, 10, 20, 40\).

12. l. 48: \(\_, 2, \_, \_, \_, 8, \_, \_, \_\)
- Factors of 48: \(1, 2, 3, 4, 6, 8, 12, 16, 24, 48\).
- Missing factors: \(1, 3, 4, 6, 12, 16, 24, 48\).
- Completed: \(1, 2, 3, 4, 6, 8, 12, 16, 24, 48\).

13. m. 72: \(\_, \_, \_, 4, \_, 8, \_, \_, \_, \_, \_\)
- Factors of 72: \(1, 2, 3, 4, 6, 8, 9, 12, 18, 24, 36, 72\).
- Missing factors: \(1, 2, 3, 6, 9, 12, 18, 24, 36, 72\).
- Completed: \(1, 2, 3, 4, 6, 8, 9, 12, 18, 24, 36, 72\).

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#### Section 4: Complete the sentences

1. a. Some numbers have exactly 2 factors. These numbers are called ___________ ___________.
- Solution: Numbers with exactly 2 factors are called prime numbers.
- Completed sentence: Some numbers have exactly 2 factors. These numbers are called prime numbers.

2. b. Most numbers have an even number of factors. The numbers that have an odd number of factors are all ___________ ___________.
- Solution: Numbers with an odd number of factors are all perfect squares. This is because in a perfect square, one of the factors is repeated (e.g., \(6^2 = 36\) has factors \(1, 2, 3, 4, 6, 9, 12, 18, 36\), where 6 is repeated).
- Completed sentence: Most numbers have an even number of factors. The numbers that have an odd number of factors are all perfect squares.

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Final Answer

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