## Questions and Answers

What is the purpose of using the Fundamental Theorem of Arithmetic in Class IX?

What can be obtained by multiplying prime numbers in the Fundamental Theorem of Arithmetic?

What is the significance of the prime factorization of the denominator in the decimal expansion of a rational number?

What is the result of multiplying some or all of the prime numbers 2, 3, 7, 11, and 23?

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What is the assumption made about the collection of primes in the Fundamental Theorem of Arithmetic?

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What is the purpose of listing the examples of multiplying prime numbers in the Fundamental Theorem of Arithmetic?

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What does Euclid's division algorithm deal with?

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What is the main purpose of using Euclid's division algorithm in this chapter?

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What is the Fundamental Theorem of Arithmetic related to?

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What is the main application of the Fundamental Theorem of Arithmetic in this chapter?

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What is the significance of the Fundamental Theorem of Arithmetic?

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What did you begin to explore in Class IX?

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What is the result of combining all primes in all possible ways?

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What is the goal of factorizing positive integers?

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What is true about the number 32760?

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What is the Fundamental Theorem of Arithmetic?

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What is the purpose of checking if 3803 and 3607 are primes?

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What is the result of expressing a number as a product of powers of primes?

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What is the definition of an irrational number?

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What is the theorem used in the proof that a prime number is irrational?

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What is the condition for a number to be called irrational?

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Which of the following numbers is an example of an irrational number?

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What is the theorem that states that if a prime number p divides a^2, then p divides a?

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What is the purpose of the proof in this section?

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What is the technique used in the proof of Theorem 1.3?

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What can be concluded about the integers r and s in the proof of Theorem 1.3?

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What is the result of squaring both sides of the equation b² = a in the proof of Theorem 1.3?

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What is the contradiction that arises in the proof of Theorem 1.3?

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What is the conclusion of the proof of Theorem 1.3?

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What is the result of squaring both sides of the equation b³ = a in the proof of Example 5?

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What is the significance of the order of prime factors in the factorisation of a composite number according to the Fundamental Theorem of Arithmetic?

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Who is credited with providing the first correct proof of the Fundamental Theorem of Arithmetic?

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What is the key feature of the factorisation of a composite number, according to the Fundamental Theorem of Arithmetic?

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What is the importance of the Fundamental Theorem of Arithmetic in number theory?

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What is the significance of Euclid's Elements in the context of the Fundamental Theorem of Arithmetic?

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What is the reputation of Carl Friedrich Gauss in the mathematical community?

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Explain the concept of an irrational number and provide an example.

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State the theorem that is used to prove that a prime number is irrational.

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What is the significance of the Fundamental Theorem of Arithmetic in the proof of the irrationality of a prime number?

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How can we prove that the square root of 2 is irrational?

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What is the relationship between the prime factors of a and a^2?

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What is the purpose of the proof of the irrationality of a prime number?

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What is the reasoning behind the conclusion that 3 is irrational in the given proof?

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What is the purpose of assuming the contrary in the proof of Example 6, i.e., that 5 – 3 is rational?

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What is the significance of the fact that a and b are integers in the proof of Example 6?

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What is the underlying assumption in the proof of Example 7, i.e., that 3 2 is rational?

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What is the relationship between the rationality of a and b, and the rationality of a² and b²?

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What is the importance of Theorem 1.3 in the context of irrational numbers?

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What is the significance of listing the prime factors in ascending order in the prime factorization of a composite number?

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How does the Fundamental Theorem of Arithmetic help in determining whether a number is divisible by 5 or not?

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What is the importance of the uniqueness of the prime factorization of a number in the Fundamental Theorem of Arithmetic?

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How does the prime factorization of a number help in determining whether it ends with a particular digit or not?

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What is the significance of the Fundamental Theorem of Arithmetic in finding the HCF and LCM of two positive integers?

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What is the relationship between the prime factorization of a number and its divisibility by other numbers?

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What is the significance of the rearrangement of the equation $3^2 = ab$ in the proof of the irrationality of $3^2$?

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What is the importance of the uniqueness of the prime factorization in the Fundamental Theorem of Arithmetic?

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How does the proof of the irrationality of $2$ lead to the conclusion that $3^2$ is irrational?

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What is the relationship between the HCF and LCM of three positive integers p, q, and r?

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What is the purpose of the exercise to prove that $3 + 2rac{1}{2}$ is irrational?

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What is the significance of the fact that the product of the HCF and LCM of three positive integers p, q, and r is not equal to p × q × r?

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