Math Case Study Class 10 Number System

Math Case Study Class 10 Number System | Free Online Test

Math Case Study Class 10 Number System

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Math case study questions class 10 encourage analytical thinking and logical reasoning. For instance, questions on rational and irrational numbers allow deeper understanding. Furthermore, solving these problems enhances critical thinking. Short exercises help reinforce key formulas. Therefore, students gain confidence and clarity in the number system through consistent practice.

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Case Study: Fundamental Theorem of Arithmetic

Case Study 3: Applying the Fundamental Theorem of Arithmetic

Ramesh is learning to apply the Fundamental Theorem of Arithmetic in real-life scenarios. He considers different numbers to check their prime factorizations. He realizes that if he knows the prime factorization, he can easily determine divisibility, HCF, and LCM. Ramesh solves several word problems involving gears, traffic lights, and schedules using LCM and HCF concepts based on prime factorization.

Key concepts:

  • The Fundamental Theorem of Arithmetic states that every integer greater than 1 can be represented uniquely as a product of prime numbers
  • Prime Factorization is the process of expressing a number as a product of its prime factors
  • Highest Common Factor (HCF) is the largest number that divides two or more numbers
  • Least Common Multiple (LCM) is the smallest number that is a multiple of two or more numbers
  • HCF and LCM can be easily calculated using prime factorization

1. Which of the following numbers is a prime number?

  • A) 49
  • B) 23
  • C) 91
  • D) 33
Answer: B) 23
Solution: 23 has only two factors: 1 and 23.

2. The HCF of 18 and 24 is:

  • A) 6
  • B) 3
  • C) 9
  • D) 12
Answer: A) 6
Solution: \(18 = 2 \times 3^2\), \(24 = 2^3 \times 3\), HCF = \(2 \times 3 = 6\)

3. Find the LCM of 18 and 24.

  • A) 36
  • B) 48
  • C) 72
  • D) 54
Answer: C) 72
Solution: LCM = \(2^3 \times 3^2 = 72\)

4. What is the prime factorization of 84?

  • A) \(2^2 \times 3 \times 7\)
  • B) \(3 \times 4 \times 7\)
  • C) \(2 \times 6 \times 7\)
  • D) \(2 \times 3 \times 14\)
Answer: A) \(2^2 \times 3 \times 7\)
Solution: \(84 = 2^2 \times 3 \times 7\)

5. Which of the following numbers is not divisible by 3?

  • A) 123
  • B) 81
  • C) 72
  • D) 74
Answer: D) 74
Solution: Sum of digits of 74 = 7 + 4 = 11, not divisible by 3.

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Prime Factorization Case Study

Case Study 1: Prime Factorization and the Fundamental Theorem of Arithmetic

Rohan is preparing a math model for his school exhibition based on the concept of prime factorization. While demonstrating the Fundamental Theorem of Arithmetic, he selects numbers such as 36, 48, and 60 and expresses them as products of prime numbers. He then tries to explore the HCF and LCM of these numbers using their prime factorizations. Rohan realizes that understanding this theorem helps in solving many real-life problems related to divisibility and factorization.

Key Concepts:

  • Prime Factorization: Expressing a number as a product of prime numbers
  • Fundamental Theorem of Arithmetic: Every integer greater than 1 can be represented uniquely as a product of prime numbers
  • HCF (Highest Common Factor): The largest number that divides two or numbers
  • LCM (Least Common Multiple): The smallest number that is a multiple of two or more numbers

1. What is the prime factorization of 60?

Answer: A) \(2 \times 2 \times 3 \times 5\)
Solution: \(60 = 2^2 \times 3 \times 5\)

2. Find the HCF of 36 and 60 using prime factorization.

Answer: B) 12
Solution: \(36 = 2^2 \times 3^2\), \(60 = 2^2 \times 3 \times 5\), HCF = \(2^2 \times 3 = 12\)

3. Find the LCM of 36 and 60 using prime factorization.

Answer: B) 360
Solution: LCM = \(2^2 \times 3^2 \times 5 = 360\)

4. Which of the following numbers is a composite number?

Answer: C) 21
Solution: 21 has more than two factors: 1, 3, 7, 21

5. If HCF(48, x) = 12 and x is a multiple of 12, what can be a possible value of x?

Answer: B) 12
Solution: \(48 = 2^4 \times 3\), \(12 = 2^2 \times 3\), HCF = \(2^2 \times 3 = 12\)

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