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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CBSE Learning Objectives – Key Concepts & Skills You Must Know

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.

CBSE Revision Notes & Quick Summary for Last-Minute Study

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

CBSE Exam Tips, Important Questions & Common Mistakes to Avoid

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.

CBSE Quiz & Practice Test – MCQs, True/False Questions with Solutions

Multiple Choice Questions

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.

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.

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.

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 HCF of two positive integers.

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

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

It is used to prove the irrationality of numbers like √2.

Correct Answer: A

Solution:

Euclid's division algorithm is primarily used to compute the HCF of two positive integers.

He is known for proving the Fundamental Theorem of Arithmetic.

He is known for inventing the Euclidean algorithm.

He is known for his work on the Pythagorean theorem.

He is known for discovering irrational numbers.

Correct Answer: A

Solution:

Carl Friedrich Gauss is credited with the first correct proof of the Fundamental Theorem of Arithmetic in his work Disquisitiones Arithmeticae.

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.

\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.

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.

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.

By determining if the number is a prime.

By analyzing the prime factorization of the denominator.

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

By converting the number into a fraction.

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.

Euclid

Isaac Newton

Carl Friedrich Gauss

Archimedes

Correct Answer: C

Solution:

Carl Friedrich Gauss is often referred to as the 'Prince of Mathematicians'.

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}

Correct Answer: C

Solution:

15 is a composite number and can be expressed as a product of primes: 3 and 5.

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.

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'.

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.

A remainder smaller than the divisor

A quotient larger than the divisor

A remainder larger than the divisor

A quotient smaller than the divisor

Correct Answer: A

Solution:

Euclid's division algorithm results in a remainder that is smaller than the divisor.

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

\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.

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

0 \leq r < b

. This is the basis for the long division process.

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

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.

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.

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.

It is a terminating decimal.

It is a non-terminating, repeating decimal.

It is an irrational number.

It is an integer.

Correct Answer: A

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.

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.

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 denominator has only the prime factors 2 and 3.

The denominator has only the prime factors 2 and 5.

The denominator has only the prime factors 3 and 5.

The denominator can have any prime factors.

Correct Answer: B

Solution:

For a rational number to have a terminating decimal expansion, its denominator (after simplifying the fraction) must have only the prime factors 2 and 5.

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

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.

To compute the HCF of two positive integers

To find the prime factorization of a number

To determine the decimal expansion of a rational number

To prove the irrationality of numbers

Correct Answer: A

Solution:

Euclid's division algorithm is primarily used to compute the HCF (Highest Common Factor) of two positive integers.

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 will be non-terminating.

It will be terminating.

It will be repeating.

It will be irrational.

Correct Answer: B

Solution:

A rational number with a denominator having only the prime factors 2 and 5 will have a terminating decimal expansion.

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 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.

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.

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.

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

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.

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.

\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.

Physics and Chemistry

Mathematics and Astronomy

Biology and Medicine

Literature and Art

Correct Answer: B

Solution:

Carl Friedrich Gauss made fundamental contributions to mathematics and science, including astronomy.

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.

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 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.

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.

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

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.

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.

\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}

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}

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

\sqrt{2}

and

Solution:

Gauss is commonly known as the 'Prince of Mathematicians' due to his significant contributions to mathematics and science.

Correct Answer: False

Solution:

An equivalent version of the theorem was first recorded in Euclid's Elements, but Gauss provided the first correct proof.

Real Numbers

AI Learning Assistant

CBSE Learning Objectives – Key Concepts & Skills You Must Know

CBSE Revision Notes & Quick Summary for Last-Minute Study

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

CBSE Exam Tips, Important Questions & Common Mistakes to Avoid

Common Mistakes and Exam Tips

Common Pitfalls

Tips for Success

Multiple Choice Questions

Q1.How does the Fundamental Theorem of Arithmetic help in determining the nature of the decimal expansion of a rational number?

Correct Answer: B

Q2.Which of the following statements is true regarding the Fundamental Theorem of Arithmetic?

Correct Answer: A

Q3.Carl Friedrich Gauss, known as the 'Prince of Mathematicians', made significant contributions to which of the following fields?

Correct Answer: A

Q4.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

Q5.Which of the following statements is true about the Euclid's division algorithm?

Correct Answer: A

Q6.Which of the following statements about Carl Friedrich Gauss is true?

Correct Answer: A

Q7.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

Q8.Which of the following numbers is irrational, as proven using the Fundamental Theorem of Arithmetic?

Correct Answer: B

Q9.Using the Fundamental Theorem of Arithmetic, which of the following numbers is a composite number?

Correct Answer: C

Q10.How does the prime factorization of the denominator of a rational number affect its decimal expansion?

Correct Answer: A

Q11.How can the Fundamental Theorem of Arithmetic help in understanding the decimal expansion of a rational number?

Correct Answer: B

Q12.Which mathematician is often referred to as the 'Prince of Mathematicians'?

Correct Answer: C

Q13.Which of the following is an application of the Fundamental Theorem of Arithmetic?

Correct Answer: A

Q14.Which of the following numbers can be expressed as a product of primes?

Correct Answer: C

Q15.According to the Fundamental Theorem of Arithmetic, which of the following is true?

Correct Answer: A

Q16.Who is credited with providing the first correct proof of the Fundamental Theorem of Arithmetic?

Correct Answer: C

Q17.What is the Fundamental Theorem of Arithmetic?

Correct Answer: B

Q18.What is the result of applying Euclid's division algorithm?

Correct Answer: A

Q19.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

Q20.Which of the following numbers is irrational?

Correct Answer: A

Q21.Which of the following statements correctly describes Euclid's division algorithm?

Correct Answer: A

Q22.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

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

Correct Answer: B

Q24.What does Euclid's division algorithm primarily deal with?

Correct Answer: B

Q25.What is the significance of Euclid's division algorithm in mathematics?

Correct Answer: D

Q26.If the prime factorization of the denominator qqq of a rational number pq\frac{p}{q}qp​ is 23×522^3 \times 5^223×52, what can be said about the decimal expansion of pq\frac{p}{q}qp​?

Correct Answer: A

Q27.What does the prime factorization of the denominator of a rational number reveal?

Correct Answer: B

Q28.Which theorem helps in understanding the decimal expansion of a rational number?

Correct Answer: B

Q29.If the decimal expansion of a rational number is terminating, which of the following must be true about the prime factorization of its denominator?

Correct Answer: B

Q30.In the context of Euclid's division algorithm, if a positive integer aaa is divided by another positive integer bbb, what can be said about the remainder rrr?

Correct Answer: B

Q31.Who is credited with the first correct proof of the Fundamental Theorem of Arithmetic?

Correct Answer: C

Q32.What is the main application of Euclid's division algorithm?