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Prime Time

Prime Time

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Co-prime Numbers
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Common Multiples and Common Factors
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Divisibility Rules
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Perfect Numbers
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Prime Factorization
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Prime Numbers and Composite Numbers
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Twin Primes
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Learning Objectives

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Learning Objectives

  • Identify common multiples and factors of given numbers.
  • Analyze the 'idli-vada' game to understand multiples of 3 and 5.
  • Determine the number of times specific phrases are said in a counting game.
  • Explore the concept of prime and composite numbers within a specified range.
  • Apply prime factorization to various numbers and identify their properties.
  • Recognize and define perfect numbers and their characteristics.
  • Solve problems related to divisibility and factors of numbers.
  • Investigate the relationships between prime numbers and their sums.

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Revision Notes & Summary

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Chapter 5 - Solutions

Prime Time

Common Multiples and Common Factors

  • Idli-Vada Game: Children sit in a circle and play a game of numbers.
    • Players say 'idli' for multiples of 3 (e.g., 3, 6, 9, 12, 18).
    • Players say 'vada' for multiples of 5 (e.g., 5, 10, 15, 20).
    • For numbers that are multiples of both 3 and 5, players say 'idli-vada' (first instance is 15).

Prime Factorization

  • Examples:
    • 64 = 2 x 2 x 2 x 2 x 2 x 2
    • 104 = 2 x 2 x 2 x 13
    • 105 = 3 x 5 x 7
    • 243 = 3 x 3 x 3 x 3 x 3
    • 320 = 2 x 2 x 2 x 2 x 2 x 5

Co-prime Numbers

  • Two numbers are co-prime if they have no common factors other than 1.

Prime and Composite Numbers

  • Prime Numbers: Numbers like 2, 3, 5, 7, 11 (only two factors: 1 and themselves).
  • Composite Numbers: Numbers like 4, 6, 8, 9 (more than two factors).

Common Mistakes

  • Confusing prime numbers with composite numbers.
  • Misidentifying common multiples.

Tips

  • Always check the prime factorization to determine if numbers are co-prime.
  • Use the idli-vada game to practice identifying multiples.
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Practice Test – MCQs, True/False

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Multiple Choice Questions

A.

18 and 35

B.

15 and 37

C.

30 and 45

D.

81 and 18
Correct Answer: A

Solution:

18 and 35 are co-prime because they have no common factors other than 1.

Chapter Concept:

Prime Factorization

A.

It is always divisible by 2.

B.

It is always divisible by 4.

C.

It is always divisible by 8.

D.

It is not divisible by any of these numbers.
Correct Answer: A

Solution:

A number ending in 8 is always divisible by 2, as it is an even number.

Chapter Concept:

Divisibility Rules

A.

9

B.

15

C.

30

D.

45
Correct Answer: B

Solution:

The smallest number that is divisible by both 3 and 5 is their least common multiple, which is 15.

Chapter Concept:

Divisibility Rules

A.

6

B.

8

C.

9

D.

10
Correct Answer: A

Solution:

A perfect number is defined as a number where the sum of all its factors equals twice the number itself. The number 6 is a perfect number because its factors are 1, 2, 3, and 6, and their sum is 12, which is twice 6.

Chapter Concept:

Perfect Numbers

A.

6

B.

8

C.

10

D.

12
Correct Answer: A

Solution:

The number 6 is a perfect number because its factors are 1, 2, 3, and 6, and their sum is 12, which is twice 6.

Chapter Concept:

Perfect Numbers

A.

120

B.

150

C.

135

D.

165
Correct Answer: B

Solution:

The 10th occurrence of 'idli-vada' happens at 150, which is a multiple of both 3 and 5.

Chapter Concept:

Common Multiples and Common Factors

A.

15 and 30 are co-prime numbers.

B.

15 is a factor of 30.

C.

30 is a factor of 15.

D.

15 and 30 have no common factors.
Correct Answer: B

Solution:

15 is a factor of 30 because 30 can be divided by 15 without leaving a remainder (30 ÷ 15 = 2). Option A is incorrect because co-prime numbers have no common factors other than 1, and 15 and 30 share common factors. Option C is incorrect because 30 is not a factor of 15. Option D is incorrect because 15 and 30 share common factors, such as 1, 3, 5, and 15.

Chapter Concept:

Prime Numbers and Composite Numbers

A.

10

B.

15

C.

20

D.

25
Correct Answer: B

Solution:

The first common multiple of 3 and 5 is 15.

Chapter Concept:

Common Multiples and Common Factors

A.

The pair is co-prime.

B.

The pair is not co-prime.

C.

The pair has a common factor greater than 1.

D.

The pair has no common multiple.
Correct Answer: B

Solution:

If the first common multiple of two numbers is less than the product of the two numbers, it indicates that the pair is not co-prime.

Chapter Concept:

Co-prime Numbers

A.

5 times

B.

6 times

C.

7 times

D.

8 times
Correct Answer: B

Solution:

'Idli-vada' will be said 6 times when the game is played up to 90.

Chapter Concept:

Common Multiples and Common Factors

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Prime Time

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True or False

Correct Answer: True

Solution:

A perfect number is defined as a number where the sum of all its factors equals twice the number. For example, the number 28 is a perfect number because its factors are 1, 2, 4, 7, 14, and 28, and their sum is 56, which is twice 28.

Chapter Concept :

Perfect Numbers

Correct Answer: False

Solution:

In the Idli-Vada game, 'idli' is said for numbers that are multiples of 3, not 5. 'Vada' is said for multiples of 5.

Chapter Concept :

Common Multiples and Common Factors

Correct Answer: True

Solution:

A perfect number is defined as a number where the sum of all its factors equals twice the number itself. For example, the number 28 is a perfect number because its factors (1, 2, 4, 7, 14, and 28) sum to 56, which is twice 28.

Chapter Concept :

Perfect Numbers

Correct Answer: True

Solution:

Co-prime numbers are defined as pairs of numbers that do not share any common factors other than 1.

Chapter Concept :

Co-prime Numbers

Correct Answer: True

Solution:

In the Idli-Vada game, 'idli-vada' is said for numbers that are multiples of both 3 and 5. The first such number is 15.

Chapter Concept :

Common Multiples and Common Factors

Correct Answer: False

Solution:

Divisibility by 4 cannot be determined by the last digit alone. The last two digits of the number must form a number divisible by 4.

Chapter Concept :

Divisibility Rules

Correct Answer: False

Solution:

The number 1 is neither a prime nor a composite number as it only has one factor, itself.

Chapter Concept :

Prime Factorization

Correct Answer: False

Solution:

In the 'idli-vada' game, players say 'idli' for numbers that are multiples of 3, not 5.

Chapter Concept :

Common Multiples and Common Factors

Correct Answer: False

Solution:

A product of prime numbers results in a composite number because it has more than two factors.

Chapter Concept :

Prime Factorization

Correct Answer: True

Solution:

The prime factorization of any composite number is unique, meaning that it can be expressed as a product of prime numbers in only one way, disregarding the order of the factors.

Chapter Concept :

Prime Factorization

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