Fundamental Theorem of Arithmetic (Statement) MCQs Quiz | Class 10

This quiz is designed for Class X students, covering the Subject Mathematics (Code 041), Unit I: Number Systems. The topic, Fundamental Theorem of Arithmetic (Statement), includes concepts like prime factorization uniqueness and the theorem’s statement. Attempt all questions and click ‘Submit Quiz’ to see your score. You can also download a detailed answer PDF.

Understanding the Fundamental Theorem of Arithmetic

The Fundamental Theorem of Arithmetic is a foundational concept in number theory, crucial for understanding the structure of natural numbers. It provides a unique way to break down composite numbers into their most basic multiplicative components: prime numbers.

Statement of the Theorem

The theorem states that every composite number can be expressed (factorized) as a product of primes, and this factorization is unique, apart from the order in which the prime factors occur.

Let’s break down this statement:

  • Composite Number: A natural number greater than 1 that is not a prime number (i.e., it has more than two factors). Examples: 4, 6, 8, 9, 10, 12, etc.
  • Product of Primes: This means a composite number can be written as multiplication of prime numbers. A prime number is a natural number greater than 1 that has no positive divisors other than 1 and itself. Examples: 2, 3, 5, 7, 11, etc.
  • Unique Factorization: This is the most critical part. It means that for any given composite number, there is only one set of prime factors (and their powers) that multiply together to form that number.
  • Apart from the order: This clarifies that while the prime factors themselves are unique, the sequence in which we write them does not change the factorization. For example, the prime factorization of 12 is 2 x 2 x 3. This is considered the same as 2 x 3 x 2 or 3 x 2 x 2. Conventionally, we write prime factors in ascending order (e.g., 22 x 3).

Prime Factorization Uniqueness

The uniqueness aspect of the theorem is what makes it so powerful. It assures us that no matter how we start factoring a composite number, we will always arrive at the same collection of prime factors. For instance:

  • To factorize 30:
    • Start with 30 = 2 x 15 = 2 x 3 x 5
    • Or start with 30 = 3 x 10 = 3 x 2 x 5
    • Or start with 30 = 5 x 6 = 5 x 2 x 3

In all cases, the prime factors are 2, 3, and 5. The set of prime factors is {2, 3, 5}, and this set is unique for the number 30. This property is fundamental to many mathematical concepts.

Applications and Importance

The Fundamental Theorem of Arithmetic is widely used in:

  1. Finding HCF (Highest Common Factor) and LCM (Lowest Common Multiple): Prime factorization is a systematic way to find the HCF and LCM of two or more positive integers.
  2. Rational Numbers: It helps in understanding why every rational number can be written in its simplest form.
  3. Cryptography: The difficulty of factoring large numbers into primes is the basis for many modern encryption methods.
  4. Proofs in Number Theory: It is a cornerstone for proving other significant theorems in number theory.

Quick Revision Points

  • Every composite number has a unique prime factorization.
  • The order of prime factors does not affect the uniqueness.
  • Prime numbers are the building blocks of all composite numbers.
  • The number 1 is neither prime nor composite, and does not have prime factors.

Extra Practice Questions

  1. What is the smallest composite number?
  2. Write the prime factorization of 140.
  3. Is 23 a prime number? Justify your answer.
  4. Find the HCF of 18 and 48 using prime factorization.
  5. Find the LCM of 12 and 18 using prime factorization.