Podcast
Questions and Answers
When is the decimal expansion of a rational number $\frac{p}{q}$ non-terminating repeating?
When is the decimal expansion of a rational number $\frac{p}{q}$ non-terminating repeating?
- When the prime factorization of $q$ contains only the prime factor 2.
- When the prime factorization of $q$ contains only the prime factor 5.
- When $p$ is a prime number.
- When the prime factorization of $q$ contains prime factors other than 2 and 5. (correct)
According to the Fundamental Theorem of Arithmetic, how can every composite number be expressed?
According to the Fundamental Theorem of Arithmetic, how can every composite number be expressed?
- As a product of prime numbers, where the factorization is unique apart from the order of the factors. (correct)
- As a sum of prime numbers.
- As a quotient of prime numbers.
- As a difference of prime numbers.
What does the uniqueness part of the Fundamental Theorem of Arithmetic imply about the prime factorization of a composite number?
What does the uniqueness part of the Fundamental Theorem of Arithmetic imply about the prime factorization of a composite number?
- A composite number can have multiple distinct prime factorizations.
- A composite number has exactly one prime factor.
- The prime factors of a composite number are unique.
- The prime factorization is unique, except for the order in which the prime factors occur. (correct)
If $p$ is a prime number that divides $a^2$, where $a$ is a positive integer, which of the following is true?
If $p$ is a prime number that divides $a^2$, where $a$ is a positive integer, which of the following is true?
If HCF$(a, b) = 12$ and LCM$(a, b) = 144$, what is the product of $a$ and $b$?
If HCF$(a, b) = 12$ and LCM$(a, b) = 144$, what is the product of $a$ and $b$?
Given that the prime factorization of 96 is $2^5 \times 3$ and the prime factorization of 404 is $2^2 \times 101$, what is the HCF (Highest Common Factor) of 96 and 404?
Given that the prime factorization of 96 is $2^5 \times 3$ and the prime factorization of 404 is $2^2 \times 101$, what is the HCF (Highest Common Factor) of 96 and 404?
How is the Fundamental Theorem of Arithmetic used when finding both the Highest Common Factor (HCF) and Least Common Multiple (LCM) of two numbers?
How is the Fundamental Theorem of Arithmetic used when finding both the Highest Common Factor (HCF) and Least Common Multiple (LCM) of two numbers?
If it is assumed that $\sqrt{2}$ is a rational number, which of the following statements represents a correct step in a proof by contradiction?
If it is assumed that $\sqrt{2}$ is a rational number, which of the following statements represents a correct step in a proof by contradiction?
Why is the statement '6 × 72 × 120 = HCF(6, 72, 120) × LCM(6, 72, 120)' incorrect?
Why is the statement '6 × 72 × 120 = HCF(6, 72, 120) × LCM(6, 72, 120)' incorrect?
What is the first step in proving that $\sqrt{3}$ is irrational using proof by contradiction?
What is the first step in proving that $\sqrt{3}$ is irrational using proof by contradiction?
Flashcards
Euclid's Division Algorithm
Euclid's Division Algorithm
States any positive integer 'a' can be divided by another positive integer 'b' leaving a remainder 'r' smaller than 'b'.
Fundamental Theorem of Arithmetic
Fundamental Theorem of Arithmetic
Every composite number can be uniquely expressed as a product of prime numbers.
Factor Tree
Factor Tree
A method to express a composite number as a product of its prime factors, visually represented as a branching tree.
HCF and LCM Relation
HCF and LCM Relation
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Irrational Number
Irrational Number
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Theorem on Prime Divisors
Theorem on Prime Divisors
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Proof by Contradiction
Proof by Contradiction
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LCM by Prime Factorization
LCM by Prime Factorization
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HCF by Prime Factorization
HCF by Prime Factorization
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Study Notes
- Real numbers are explored, including properties of positive integers like Euclid's division algorithm, Fundamental Theorem of Arithmetic, and irrational numbers
Euclid's Division Algorithm
- Applies to the divisibility of integers
- A positive integer a can be divided by another positive integer b, leaving a remainder r that is smaller than b
- Used to compute the HCF (Highest Common Factor) of two positive integers
Fundamental Theorem of Arithmetic
- Deals with the multiplication of positive integers
- Every composite number can be expressed as a product of primes uniquely
- Used to prove the irrationality of numbers like √2, √3, and √5
- Used to explore when the decimal expansion of a rational number p/q is terminating or non-terminating repeating, based on the prime factorization of the denominator q
Prime Numbers and Factorization
- Any natural number can be written as a product of its prime factors
- A collection of prime numbers can be multiplied to produce a large collection of positive integers
- There are infinitely many prime numbers
- All composite numbers can be written as the product of powers of primes
- Example factorization: 32760 = 2³ × 3² × 5 × 7 × 13
- Theorem 1.1 (Fundamental Theorem of Arithmetic): Every composite number can be factorized as a product of primes, and this factorization is unique, except for the order of the prime factors
- Carl Friedrich Gauss (1777-1855) provided the first correct proof of the Fundamental Theorem of Arithmetic
- A composite number can be factorized as a product of prime numbers in a unique way, disregarding the order of primes
HCF and LCM
- HCF(6, 20) = 2¹ = Product of the smallest power of each common prime factor in the numbers
- LCM (6, 20) = 2² × 3¹ × 5¹ = Product of the greatest power of each prime factor involved in the numbers
- HCF (a, b) × LCM (a, b) = a × b for any two positive integers a and b
- Example: for 96 = 2⁵ × 3 and 404 = 2² × 101, HCF(96, 404) = 2² = 4
- LCM(96, 404) = (96 × 404) / HCF(96, 404) = 9696
Irrational Numbers
- A number 's' is irrational if it cannot be written in the form p/q, where p and q are integers and q ≠ 0
- Theorem 1.2:* If p is a prime number and p divides a², then p divides a, where a is a positive integer
- Assuming to the contrary which is called 'proof by contradiction'
- Used to prove that √2 is irrational
Proving Irrationality
- Assume √2 = r/s, where r and s are coprime
- Leads to a contradiction, as both a and b would have a common factor of 2
- The sum or difference of a rational and an irrational number is irrational
- The product and quotient of a non-zero rational and irrational number is irrational
- Example: To show 5 - √3 is irrational, assume 5 - √3 = a/b, where a and b are coprime
Summary of Key Points
- Every composite number can be factorized as a product of primes uniquely
- If p is prime & divides a², then p divides a
- √2 and √3 are irrational numbers
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