If $p = 7$ and $q = 125$, what is the nature of the decimal expansion of the rational number $\frac{p}{q}$?

Correct Option: A. Solution: The prime factorization of $q = 125$ is $5^3$. Since $q$ has only 5 as its prime factor, the decimal expansion of $\frac{p}{q}$ is terminating.

Given the integers $a = 56$ and $b = 15$, use the Euclid's division algorithm to find the greatest common divisor (GCD) of $a$ and $b$. What is the value of the GCD?

Correct Option: C. Solution: Using Euclid's division algorithm, we perform the division: $56 = 15 \times 3 + 11$. Next, we divide $15$ by $11$: $15 = 11 \times 1 + 4$. Then, divide $11$ by $4$: $11 = 4 \times 2 + 3$. Continue with $4$ and $3$: $4 = 3 \times 1 + 1$. Finally, $3 = 1 \times 3 + 0$. The remainder is now $0$, and the last non-zero remainder is $1$. Therefore, the GCD is $1$.

Real Numbers

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Summary

Summary of Real Numbers Chapter

Introduction to Real Numbers
- Exploration of irrational numbers continues.
- Key topics: Euclid's division algorithm and Fundamental Theorem of Arithmetic.
Euclid's Division Algorithm
- Any positive integer a can be divided by another positive integer b, leaving a remainder r smaller than b.
- Applications include computing the HCF of two positive integers.
Fundamental Theorem of Arithmetic
- Every composite number can be expressed as a unique product of prime factors.
- Applications:
  - Proving the irrationality of numbers like √2, √3, and √5.
  - Determining the nature of decimal expansions of rational numbers based on the prime factorization of the denominator.
Key Points Studied
1. Fundamental Theorem of Arithmetic: Unique factorization of composite numbers.
2. If p is prime and p divides a², then p divides a (where a is a positive integer).
3. Proving that √2 and √3 are irrational.
Important Relationships
- HCF and LCM relationships:
  - HCF(p, q, r) × LCM(p, q, r) ≠ p × q × r
  - LCM(p, q, r) = HCF(q) × HCF(q, r) × HCF(p)
  - HCF(p, q, r) = LCM(p, q) × LCM(q, r) × LCM(p, r)
Examples
- HCF and LCM calculations using prime factorization method.

Learning Objectives

Understand the Fundamental Theorem of Arithmetic.
Apply Euclid's division algorithm to find the HCF of integers.
Prove the irrationality of numbers such as √2, √3, and √5.
Explore the relationship between prime factorization and the nature of decimal expansions of rational numbers.
Calculate the HCF and LCM of given integers using prime factorization.

Detailed Notes

Chapter Notes on Real Numbers

1.1 Introduction

Exploration of real numbers and irrational numbers.
Key topics: Euclid's division algorithm and the Fundamental Theorem of Arithmetic.

1.2 The Fundamental Theorem of Arithmetic

Every composite number can be expressed as a product of primes uniquely.
Example:
- 2 = 2
- 4 = 2 x 2
- 253 = 11 x 23
Applications include:
- Proving the irrationality of numbers like √2, √3, and √5.
- Determining the nature of decimal expansions of rational numbers based on the prime factorization of their denominators.

Example of Prime Factorization

Factorization of 32760:
- 32760 = 2 x 2 x 2 x 3 x 3 x 5 x 7 x 13 = 2² x 3² x 5 x 7 x 13

1.3 Revisiting Irrational Numbers

Definition: A number is irrational if it cannot be expressed as p/q where p and q are integers and q ≠ 0.
Examples of irrational numbers: √2, √3, √15, π.
Theorem 1.2: If p is a prime and p divides a², then p divides a.

Proof of Irrationality of √2

Assume √2 is rational, leading to a contradiction.

1.4 Summary

The Fundamental Theorem of Arithmetic states that every composite number can be uniquely factorized into primes.
If p is a prime and p divides a², then p divides a.
Proved that √2 and √3 are irrational.

Important Formulas

Formula	Description
HCF(p, q, r) x LCM(p, q, r) = p x q x r	Relationship between HCF and LCM of three numbers
LCM(p, q, r) = HCF(q) x HCF(q, r) x HCF(p)	LCM in terms of HCFs
HCF(p, q, r) = LCM(p, q) x LCM(q, r) x LCM(p, r)	HCF in terms of LCMs

Exam Tips & Common Mistakes

Common Mistakes and Exam Tips

Common Pitfalls

Misunderstanding the Fundamental Theorem of Arithmetic: Students often confuse the uniqueness of prime factorization with the ability to factorize any number. Remember, every composite number can be expressed as a product of primes uniquely, except for the order.
Assuming irrational numbers can be expressed as fractions: Many students mistakenly believe that all numbers can be expressed in the form p/q. This is not true for irrational numbers like √2, √3, etc.
Incorrectly applying the HCF and LCM relationship: Students sometimes forget that the product of HCF and LCM of two numbers equals the product of the numbers themselves. This can lead to errors in calculations.

Tips for Success

Practice Prime Factorization: Regularly practice breaking down numbers into their prime factors to strengthen your understanding of the Fundamental Theorem of Arithmetic.
Understand Proofs of Irrationality: Familiarize yourself with the proofs that demonstrate the irrationality of numbers like √2 and √3. Understanding the logic behind these proofs can help avoid misconceptions.
Use Factor Trees: When factorizing larger numbers, use factor trees to visually organize your work. This can help prevent mistakes in identifying prime factors.
Check Your Work: Always verify your calculations for HCF and LCM by ensuring that they satisfy the relationship with the original numbers.

Practice & Assessment

Multiple Choice Questions

Whether the decimal expansion is terminating or non-terminating repeating.

The exact value of the rational number.

The speed at which the number converges to zero.

The maximum value the number can reach.

Correct Answer: A

Solution:

The prime factorization of the denominator

q

reveals whether the decimal expansion of

\frac{p}{q}

is terminating or non-terminating repeating.

Correct Answer: C

Solution:

The number 49 is a composite number because it can be expressed as

7 \times 7

, whereas the other options are prime numbers.

Correct Answer: A

Solution:

According to Euclid's division algorithm,

a = bq + r

where

0 \leq r < b

. Dividing 56 by 12 gives 56 = 12 \times 4 + 8. Thus, the remainder

r

is 8.

Finding the HCF of two numbers.

Proving the irrationality of numbers like √2.

Determining the greatest common divisor.

Calculating the square root of a number.

Correct Answer: B

Solution:

The Fundamental Theorem of Arithmetic is used to prove the irrationality of numbers like √2.

Mathematics and Science

Literature and Arts

Philosophy and Economics

Biology and Chemistry

Correct Answer: A

Solution:

Carl Friedrich Gauss is renowned for his fundamental contributions to both mathematics and science, making him one of the greatest mathematicians of all time.

2 \times 3 \times 5 \times 7

2^2 \times 3 \times 5

2 \times 3^2 \times 5

2 \times 3 \times 5 \times 11

Correct Answer: A

Solution:

The number

210

can be factorized as follows:

210 = 2 \times 105 = 2 \times 3 \times 35 = 2 \times 3 \times 5 \times 7

. Therefore, the prime factorization of

210

2 \times 3 \times 5 \times 7

It helps in understanding the multiplication of integers.

It is used to explore the decimal expansion of rational numbers.

It is used to prove the irrationality of numbers.

It has applications related to the divisibility properties of integers.

Correct Answer: D

Solution:

Euclid's division algorithm has many applications related to the divisibility properties of integers.

Correct Answer: A

Solution:

According to Euclid's division algorithm,

a = bq + r

where

0 \leq r < b

. Dividing 56 by 15 gives a quotient of 3 and a remainder of 11. Thus, the remainder is 11.

Every positive integer can be expressed as a sum of primes.

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

Every integer can be divided by another integer without a remainder.

Every rational number has a terminating decimal expansion.

Correct Answer: B

Solution:

The Fundamental Theorem of Arithmetic states that every composite number can be expressed as a product of primes in a unique way.

r = b

r = 0

r = a

r > b

Correct Answer: B

Solution:

According to Euclid's division algorithm, the remainder

r

when

a

is divided by

b

must satisfy

0 \leq r < b

. Therefore,

r

can be 0 but cannot be equal to or greater than

b

Euclid's division algorithm

Fundamental Theorem of Arithmetic

Pythagorean Theorem

Binomial Theorem

Correct Answer: B

Solution:

The Fundamental Theorem of Arithmetic helps in understanding when the decimal expansion of a rational number is terminating or non-terminating repeating.

The number of digits in the decimal expansion.

Whether the decimal expansion is terminating or non-terminating repeating.

The sum of the digits in the decimal expansion.

The square root of the rational number.

Correct Answer: B

Solution:

The prime factorization of the denominator of a rational number reveals whether its decimal expansion is terminating or non-terminating repeating.

\sqrt{2}

0.75

Correct Answer: A

Solution:

\sqrt{2}

is an example of an irrational number, as it cannot be expressed as a fraction of two integers.

Multiplication of integers

Divisibility of integers

Addition of integers

Subtraction of integers

Correct Answer: B

Solution:

Euclid's division algorithm deals with the divisibility of integers, allowing one integer to be divided by another with a remainder.

Every integer greater than 1 is either a prime number or can be uniquely factorized into prime numbers.

Every integer can be factorized into prime numbers, but the factorization is not unique.

Prime numbers can be factorized into composite numbers.

The theorem only applies to even numbers.

Correct Answer: A

Solution:

The Fundamental Theorem of Arithmetic states that every integer greater than 1 is either a prime number or can be uniquely expressed as a product of prime numbers, up to the order of the factors.

The decimal expansion is always terminating.

The decimal expansion is always non-terminating and non-repeating.

The decimal expansion is terminating if the prime factorization of

q

contains only the primes 2 and/or 5.

The decimal expansion is non-terminating repeating if

q

is a prime number.

Correct Answer: C

Solution:

The Fundamental Theorem of Arithmetic implies that the decimal expansion of a rational number

\frac{p}{q}

is terminating if the prime factorization of

q

contains only the primes 2 and/or 5.

It determines whether the expansion is terminating or non-terminating repeating.

It determines whether the number is rational or irrational.

It affects the ability to find the square root of the number.

It affects the ability to find the HCF of the number.

Correct Answer: A

Solution:

The prime factorization of the denominator of a rational number reveals whether its decimal expansion is terminating or non-terminating repeating.

Euclid

Carl Friedrich Gauss

Isaac Newton

Archimedes

Correct Answer: B

Solution:

Carl Friedrich Gauss is often referred to as the 'Prince of Mathematicians' due to his significant contributions to mathematics and science.

It helps in finding the square root of decimal numbers.

It determines when the decimal expansion of a rational number is terminating.

It is used to convert decimals to fractions.

It helps in rounding off decimal numbers.

Correct Answer: B

Solution:

The Fundamental Theorem of Arithmetic can be used to determine when the decimal expansion of a rational number is terminating by examining the prime factorization of the denominator.

It allows for the addition of prime numbers

It states that every composite number can be expressed uniquely as a product of primes

It helps in finding the least common multiple

It is used for subtracting prime numbers

Correct Answer: B

Solution:

The Fundamental Theorem of Arithmetic states that every composite number can be expressed as a product of primes in a unique way.

Multiplication of integers.

Divisibility of integers.

Prime factorization.

Decimal expansion of rational numbers.

Correct Answer: B

Solution:

Euclid's division algorithm primarily deals with the divisibility of integers.

To calculate the square root of a number.

To prove the irrationality of numbers like

\sqrt{2}

and

\sqrt{3}

To find the greatest common divisor of two numbers.

To solve quadratic equations.

Correct Answer: B

Solution:

One application of the Fundamental Theorem of Arithmetic is to prove the irrationality of numbers like

\sqrt{2}

and

\sqrt{3}

It states that every integer can be expressed as a sum of primes.

It states that every composite number can be expressed as a product of primes in a unique way.

It is used to calculate the greatest common divisor of two numbers.

It is used to determine the divisibility of integers.

Correct Answer: B

Solution:

The Fundamental Theorem of Arithmetic states that every composite number can be expressed as a product of primes in a unique way.

The remainder

r

is always greater than

b

The remainder

r

is always less than

b

The remainder

r

can be equal to

b

The remainder

r

is always equal to zero.

Correct Answer: B

Solution:

Euclid's division algorithm states that for any two positive integers

a

and

b

, there exist unique integers

q

and

r

such that

a = bq + r

where

0 \leq r < b

. Therefore, the remainder

r

is always less than

b

The remainder

r

must be greater than

b

The remainder

r

must be less than or equal to

b

The remainder

r

must be less than

b

The remainder

r

must be equal to

b

Correct Answer: C

Solution:

According to Euclid's division algorithm, for two positive integers

a

and

b

, the remainder

r

when

a

is divided by

b

must be less than

b

\sqrt{2}

\sqrt{4}

\sqrt{9}

\sqrt{16}

Correct Answer: A

Solution:

The Fundamental Theorem of Arithmetic is used to prove the irrationality of numbers like

\sqrt{2}

Correct Answer: D

Solution:

Using Euclid's division algorithm,

56 \div 9

gives a quotient of 6 and a remainder of 7.

It is used to find the remainder when one integer is divided by another.

It is used to express a composite number as a product of primes.

It is used to determine if a number is irrational.

It is used to find the decimal expansion of a rational number.

Correct Answer: A

Solution:

Euclid's division algorithm is used to find the remainder when one integer is divided by another, which is a key part of the long division process.

Proving the irrationality of numbers like

\sqrt{2}

and

\sqrt{3}

Solving quadratic equations.

Calculating the area of a circle.

Determining the speed of light.

Correct Answer: A

Solution:

One application of the Fundamental Theorem of Arithmetic is proving the irrationality of numbers like

\sqrt{2}

and

\sqrt{3}

It states that any positive integer

a

can be divided by another positive integer

b

to leave a remainder

r

less than

b

It is used to express composite numbers as a product of primes.

It determines whether a rational number has a terminating decimal expansion.

It is a method for finding the least common multiple of two numbers.

Correct Answer: A

Solution:

Euclid's division algorithm states that for any two positive integers

a

and

b

, there exist unique integers

q

and

r

such that

a = bq + r

, where

Solution:

A rational number

\frac{p}{q}

has a terminating decimal expansion if the prime factorization of

q

contains only the primes 2 and 5. Here,

q = 2^3 \times 5^2

satisfies this condition.

They can be divided by any integer.

They can be expressed as a product of primes in a unique way.

They are always even numbers.

They have a remainder of zero when divided by any integer.

Correct Answer: B

Solution:

The Fundamental Theorem of Arithmetic states that every composite number can be expressed as a product of primes in a unique way.

By expressing the numerator as a product of primes.

By analyzing the prime factorization of the denominator.

By converting the rational number into a fraction with a prime denominator.

By determining the greatest common divisor of the numerator and denominator.

Correct Answer: B

Solution:

The Fundamental Theorem of Arithmetic helps in determining the nature of the decimal expansion of a rational number by analyzing the prime factorization of the denominator. If the denominator, after simplification, has no prime factors other than 2 or 5, the decimal expansion is terminating.

Euclid

Isaac Newton

Carl Friedrich Gauss

Archimedes

Correct Answer: C

Solution:

Carl Friedrich Gauss provided the first correct proof of the Fundamental Theorem of Arithmetic in his work 'Disquisitiones Arithmeticae'.

The divisibility of integers.

The multiplication of positive integers.

The addition of irrational numbers.

The subtraction of rational numbers.

Correct Answer: B

Solution:

The Fundamental Theorem of Arithmetic deals with the multiplication of positive integers, stating that every composite number can be expressed as a product of primes in a unique way.

\sqrt{4}

\sqrt{9}

\sqrt{2}

\sqrt{16}

Correct Answer: C

Solution:

The number

\sqrt{2}

is irrational because it cannot be expressed as a ratio of two integers. The other options,

\sqrt{4}

\sqrt{9}

, and

\sqrt{16}

, are rational because they are equal to 2, 3, and 4 respectively.

Terminating

Non-terminating repeating

Non-terminating non-repeating

Cannot be determined

Correct Answer: A

Solution:

The prime factorization of

q = 125

5^3

. Since

q

has only 5 as its prime factor, the decimal expansion of

\frac{p}{q}

is terminating.

Correct Answer: C

Solution:

Using Euclid's division algorithm, we perform the division:

56 = 15 \times 3 + 11

. Next, we divide

15

11

15 = 11 \times 1 + 4

. Then, divide

11

4

11 = 4 \times 2 + 3

. Continue with

4

and

3

4 = 3 \times 1 + 1

. Finally,

3 = 1 \times 3 + 0

. The remainder is now

0

, and the last non-zero remainder is

1

. Therefore, the GCD is

1

Every integer greater than 1 is either a prime number or can be uniquely expressed as a product of prime numbers.

Every integer can be expressed as a sum of prime numbers.

Every integer has a unique decimal representation.

Every integer can be expressed as a product of even numbers.

Correct Answer: A

Solution:

The Fundamental Theorem of Arithmetic states that every integer greater than 1 is either a prime number or can be uniquely expressed as a product of prime numbers, up to the order of the factors.

r

is always greater than

b

r

is always less than

b

r

is equal to

b

r

is always zero.

Correct Answer: B

Solution:

According to Euclid's division algorithm, the remainder

r

is always less than the divisor

b

\sqrt{4}

\sqrt{6}

\sqrt{9}

\sqrt{16}

Correct Answer: B

Solution:

The number

\sqrt{6}

is irrational because it cannot be expressed as a fraction of two integers, which can be shown using the Fundamental Theorem of Arithmetic. In contrast,

\sqrt{4}

\sqrt{9}

, and

\sqrt{16}

are integers.

Euclid

Solution:

The Fundamental Theorem of Arithmetic is used to prove the irrationality of numbers such as

\sqrt{2}

and

Solution:

The Fundamental Theorem of Arithmetic states that every composite number can be expressed as a product of primes in a unique way.

Real Numbers

AI Learning Assistant

Summary

Summary of Real Numbers Chapter

Learning Objectives

Detailed Notes

Chapter Notes on Real Numbers

1.1 Introduction

1.2 The Fundamental Theorem of Arithmetic

Example of Prime Factorization

1.3 Revisiting Irrational Numbers

Proof of Irrationality of √2

1.4 Summary

Important Formulas

Exam Tips & Common Mistakes

Common Mistakes and Exam Tips

Common Pitfalls

Tips for Success

Practice & Assessment

Multiple Choice Questions

Q1.What does the prime factorization of the denominator qqq of a rational number pq\frac{p}{q}qp​ reveal about its decimal expansion?

Correct Answer: A

Q2.Using the Fundamental Theorem of Arithmetic, which of the following numbers is a composite number?

Correct Answer: C

Q3.Consider two positive integers a=56a = 56a=56 and b=12b = 12b=12. Using Euclid's division algorithm, what is the remainder rrr when aaa is divided by bbb?

Correct Answer: A

Q4.Which application of the Fundamental Theorem of Arithmetic is mentioned in the excerpt?

Correct Answer: B

Q5.Carl Friedrich Gauss, known as the 'Prince of Mathematicians', made significant contributions to which of the following fields?

Correct Answer: A

Q6.Consider the number n=210n = 210n=210. According to the Fundamental Theorem of Arithmetic, what is the prime factorization of nnn?

Correct Answer: A

Q7.What is the significance of Euclid's division algorithm in mathematics?

Correct Answer: D

Q8.Consider two positive integers a=56a = 56a=56 and b=15b = 15b=15. Using Euclid's division algorithm, what is the remainder when aaa is divided by bbb?

Correct Answer: A

Q9.What is the Fundamental Theorem of Arithmetic?

Correct Answer: B

Q10.Consider two positive integers aaa and bbb where a>ba > ba>b. According to Euclid's division algorithm, which of the following is a possible value for the remainder rrr when aaa is divided by bbb?

Correct Answer: B

Q11.Which theorem helps in understanding the decimal expansion of a rational number?

Correct Answer: B

Q12.What does the prime factorization of the denominator of a rational number reveal?

Correct Answer: B

Q13.Which of the following numbers is irrational?

Correct Answer: A

Q14.What does Euclid's division algorithm primarily deal with?

Correct Answer: B

Q15.Which of the following statements is true regarding the Fundamental Theorem of Arithmetic?

Correct Answer: A

Q16.What does the Fundamental Theorem of Arithmetic imply about the decimal expansion of a rational number pq\frac{p}{q}qp​?

Correct Answer: C

Q17.How does the prime factorization of the denominator of a rational number affect its decimal expansion?

Correct Answer: A

Q18.Which mathematician is often referred to as the 'Prince of Mathematicians'?

Correct Answer: B

Q19.How can the Fundamental Theorem of Arithmetic be used in relation to decimal expansions?

Correct Answer: B

Q20.What is the significance of the Fundamental Theorem of Arithmetic in terms of prime factorization?

Correct Answer: B

Q21.What is the main focus of Euclid's division algorithm?

Correct Answer: B

Q22.What is one application of the Fundamental Theorem of Arithmetic?

Correct Answer: B

Q23.Which of the following statements is true about the Fundamental Theorem of Arithmetic?

Correct Answer: B

Q24.Consider two positive integers aaa and bbb where a>ba > ba>b. According to Euclid's division algorithm, which of the following statements is true?

Correct Answer: B

Q25.According to Euclid's division algorithm, if aaa and bbb are positive integers, which condition must be satisfied for the remainder rrr?

Correct Answer: C

Q26.Which of the following numbers is used to prove its irrationality using the Fundamental Theorem of Arithmetic?

Correct Answer: A

Q27.If a=56a = 56a=56 and b=9b = 9b=9, what is the remainder when aaa is divided by bbb using Euclid's division algorithm?

Correct Answer: D

Q28.Which of the following best describes Euclid's division algorithm?

Correct Answer: A

Q29.Which of the following is an application of the Fundamental Theorem of Arithmetic?

Correct Answer: A

Q30.Which of the following statements correctly describes Euclid's division algorithm?

Correct Answer: A