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Questions and Answers
What is the HCF of the numbers 6 and 20?
What is the HCF of the numbers 6 and 20?
What is the LCM of the numbers 6 and 20?
What is the LCM of the numbers 6 and 20?
Which statement about the relationship between HCF and LCM is correct?
Which statement about the relationship between HCF and LCM is correct?
Using the prime factorization method, what is the prime factorization of 96?
Using the prime factorization method, what is the prime factorization of 96?
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How would you find the LCM of 96 and 404 using their HCF?
How would you find the LCM of 96 and 404 using their HCF?
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What is the HCF of the numbers 6, 72, and 120?
What is the HCF of the numbers 6, 72, and 120?
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Which of the following is true regarding the product of three numbers and their HCF and LCM?
Which of the following is true regarding the product of three numbers and their HCF and LCM?
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The prime factorization of 404 is represented as?
The prime factorization of 404 is represented as?
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Who is credited with the first correct proof of the Fundamental Theorem of Arithmetic?
Who is credited with the first correct proof of the Fundamental Theorem of Arithmetic?
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What does the Fundamental Theorem of Arithmetic state about composite numbers?
What does the Fundamental Theorem of Arithmetic state about composite numbers?
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In what manner is the prime factorisation expressed according to the Fundamental Theorem of Arithmetic?
In what manner is the prime factorisation expressed according to the Fundamental Theorem of Arithmetic?
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What would the prime factorisation of 32760 look like if expressed in powers of primes?
What would the prime factorisation of 32760 look like if expressed in powers of primes?
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Which of the following statements is true regarding the uniqueness of prime factorisation?
Which of the following statements is true regarding the uniqueness of prime factorisation?
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If a number can be expressed as 4n, what can be concluded about its prime factors?
If a number can be expressed as 4n, what can be concluded about its prime factors?
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Which mathematician is commonly referred to as the ‘Prince of Mathematicians’?
Which mathematician is commonly referred to as the ‘Prince of Mathematicians’?
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What major implication does the Fundamental Theorem of Arithmetic have on composite numbers?
What major implication does the Fundamental Theorem of Arithmetic have on composite numbers?
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What does the Fundamental Theorem of Arithmetic state regarding prime factors of a number?
What does the Fundamental Theorem of Arithmetic state regarding prime factors of a number?
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What assumption leads to a contradiction in the proof of the irrationality of √2?
What assumption leads to a contradiction in the proof of the irrationality of √2?
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In the proof that √2 is irrational, which step is crucial after establishing that 2 divides a²?
In the proof that √2 is irrational, which step is crucial after establishing that 2 divides a²?
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What is the prime factorization of 140?
What is the prime factorization of 140?
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What conclusion is made about integers a and b in the proof regarding √3 being irrational?
What conclusion is made about integers a and b in the proof regarding √3 being irrational?
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For the integers 26 and 91, what are the LCM and HCF?
For the integers 26 and 91, what are the LCM and HCF?
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What can be inferred if b² is found to be divisible by 3 in the proof for √3's rationality?
What can be inferred if b² is found to be divisible by 3 in the proof for √3's rationality?
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When using the prime factorization method, what are the prime factors of 12, 15, and 21?
When using the prime factorization method, what are the prime factors of 12, 15, and 21?
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What method is used to prove that √2 is irrational?
What method is used to prove that √2 is irrational?
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What can be concluded about the product of LCM and HCF from two numbers?
What can be concluded about the product of LCM and HCF from two numbers?
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Why is it important to express integers a and b as coprime in the context of these proofs?
Why is it important to express integers a and b as coprime in the context of these proofs?
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What is a key outcome of the proof that both a and b must be divisible by 2?
What is a key outcome of the proof that both a and b must be divisible by 2?
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Why is the number 5 irrational?
Why is the number 5 irrational?
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Given HCF(306, 657) = 9, what is LCM(306, 657)?
Given HCF(306, 657) = 9, what is LCM(306, 657)?
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If Sonia takes 18 minutes and Ravi takes 12 minutes to complete a circular path, after how many minutes will they meet again at the starting point?
If Sonia takes 18 minutes and Ravi takes 12 minutes to complete a circular path, after how many minutes will they meet again at the starting point?
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Which of the following claims is true regarding the composite numbers mentioned?
Which of the following claims is true regarding the composite numbers mentioned?
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What does Euclid's division algorithm state about dividing a positive integer?
What does Euclid's division algorithm state about dividing a positive integer?
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What is a unique characteristic of composite numbers according to the Fundamental Theorem of Arithmetic?
What is a unique characteristic of composite numbers according to the Fundamental Theorem of Arithmetic?
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How can the Fundamental Theorem of Arithmetic help determine the nature of a decimal expansion?
How can the Fundamental Theorem of Arithmetic help determine the nature of a decimal expansion?
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Which of the following is NOT an application of the Fundamental Theorem of Arithmetic?
Which of the following is NOT an application of the Fundamental Theorem of Arithmetic?
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What type of numbers were previously studied in Class IX that relate to the Fundamental Theorem of Arithmetic?
What type of numbers were previously studied in Class IX that relate to the Fundamental Theorem of Arithmetic?
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What does the remainder represent in the context of Euclid's division algorithm?
What does the remainder represent in the context of Euclid's division algorithm?
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Which expression satisfies the condition of Euclid's division algorithm for positive integers?
Which expression satisfies the condition of Euclid's division algorithm for positive integers?
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When is the decimal expansion of a rational number non-terminating repeating?
When is the decimal expansion of a rational number non-terminating repeating?
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Study Notes
Fundamental Theorem of Arithmetic
- Every composite (non-prime) number can be uniquely factored into a product of prime numbers, ignoring the order of the factors.
- Example: 32760 = 2³ × 3² × 5 × 7 × 13
- Can be used to find the Highest Common Factor (HCF) and Least Common Multiple (LCM) of two or more integers.
Euclid's Division Algorithm
- Every positive integer a can be divided by any positive integer b to obtain a remainder r, which is smaller than b.
- It can be used to compute the HCF of two positive integers.
Irrational Numbers
- A number s is irrational if it cannot be written as p/q, where p and q are integers and q ≠ 0.
- √2, √3, √5, and √p (where p is a prime) are irrational numbers.
- Proof by contradiction is used to demonstrate the irrationality of these numbers.
Applications of the Fundamental Theorem of Arithmetic
- Proving the irrationality of numbers like √2, √3, and √5.
- Determining whether the decimal expansion of a rational number is terminating or non-terminating repeating.
- The prime factorisation of the denominator of a rational number reveals the nature of its decimal expansion.
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Description
Explore the Fundamental Theorem of Arithmetic, Euclid's Division Algorithm, and the nature of irrational numbers. This quiz covers unique factorization of composite numbers, computing highest common factors, and the proof of irrationality. Test your understanding and ability to apply these concepts in various mathematical scenarios.
