Properties of Real Numbers

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What is the result of Euclid’s division algorithm?

It says any positive integer $a$ cannot be divided by another positive integer $b$.

It says any positive integer $a$ can be divided by another positive integer $b$ in such a way that it leaves a remainder $r$ equal to $b$.

It says any positive integer $a$ can be divided by another positive integer $b$ in such a way that it leaves a remainder $r$ that is smaller than $b$. (correct)

It says any positive integer $a$ can be divided by another positive integer $b$ in such a way that it leaves a remainder $r$ that is greater than $b$.

What is the Fundamental Theorem of Arithmetic?

Every composite number can be expressed as a sum of primes in a unique way.

Every prime number can be expressed as a sum of composites in a unique way.

Every composite number can be expressed as a product of primes in a unique way. (correct)

Every prime number can be expressed as a product of composites in a unique way.

What is the main application of the Fundamental Theorem of Arithmetic?

Computing the prime factorization of composite numbers. (correct)

Finding the greatest common divisor (GCD) of two numbers.

Solving quadratic equations.

Computing the least common multiple (LCM) of two numbers.

What does the Fundamental Theorem of Arithmetic say about composite numbers?

They can be expressed as a product of primes in a unique way. Signup and view all the answers

How does Euclid’s division algorithm relate to finding the greatest common divisor (GCD) of two numbers?

$ ext{GCD}(a,b)$ can be computed using the remainders obtained when $a$ is divided by $b$ using Euclid’s division algorithm. Signup and view all the answers

What is the main application of Euclid’s division algorithm?

Computing the highest common factor (HCF) of two positive integers Signup and view all the answers

What is the significance of the Fundamental Theorem of Arithmetic?

It states that every composite number can be expressed as a product of primes in a unique way Signup and view all the answers

In what way does Euclid’s division algorithm relate to finding the greatest common divisor (GCD) of two numbers?

It is used to compute the GCD by repeatedly applying the division algorithm Signup and view all the answers

What does the Fundamental Theorem of Arithmetic state about prime factorization?

$Every$ $composite$ $number$ $can$ $be$ $expressed$ $as$ $a$ $product$ $of$ $primes$ $in$ $a$ $unique$ $way$ Signup and view all the answers

How does Euclid’s division algorithm differ from the Fundamental Theorem of Arithmetic?

Euclid’s division algorithm deals with divisibility of integers, while the Fundamental Theorem deals with multiplication of positive integers Signup and view all the answers

What is the significance of the Euclid’s division algorithm and the Fundamental Theorem of Arithmetic in the field of mathematics?

The Euclid’s division algorithm and the Fundamental Theorem of Arithmetic are significant in the field of mathematics as they provide foundational concepts for divisibility, prime factorization, and the computation of highest common factors and least common multiples. These concepts form the basis for many advanced mathematical ideas and applications, including cryptography and number theory. Signup and view all the answers

How does the Euclid’s division algorithm relate to the computation of the highest common factor (HCF) of two positive integers?

The Euclid’s division algorithm is utilized to compute the highest common factor (HCF) of two positive integers by repeatedly applying the algorithm to find the remainder until a remainder of 0 is obtained. The divisor at this step gives the HCF of the two numbers. Signup and view all the answers

Explain the concept of the Fundamental Theorem of Arithmetic and its uniqueness in expressing composite numbers as a product of primes.

The Fundamental Theorem of Arithmetic states that every composite number can be expressed as a product of primes in a unique way. This uniqueness is significant as it forms the basis for prime factorization, which has applications in cryptography and number theory, and is fundamental for understanding the properties of integers. Signup and view all the answers

What are the applications of the Euclid’s division algorithm and the Fundamental Theorem of Arithmetic in mathematics?

The applications of the Euclid’s division algorithm and the Fundamental Theorem of Arithmetic in mathematics include computing highest common factors and least common multiples, prime factorization, and understanding the properties of integers. Additionally, these concepts have applications in cryptography, number theory, and various areas of advanced mathematics. Signup and view all the answers

How does the Euclid’s division algorithm differ from the Fundamental Theorem of Arithmetic in terms of their applications?

The Euclid’s division algorithm primarily deals with the divisibility and computation of highest common factors, while the Fundamental Theorem of Arithmetic is focused on prime factorization and the unique representation of composite numbers as a product of primes. Their applications include different aspects of number theory and mathematical computations. Signup and view all the answers

Properties of Real Numbers

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