Question 1

If $p$ is a prime number and $p$ divides $a^2$, which of the following is a valid application of this theorem?

Accepted Answer

Correct Option: A. Solution: The theorem applies correctly in option a: since $3$ divides $9$, $a$ must be divisible by $3$.

Question 2

Assume $p$ is a prime number and $p$ divides $a^2$. If $a = 2c$ for some integer $c$, what can be inferred about $c$?

Accepted Answer

Correct Option: A. Solution: Since $p$ divides $a^2$ and $a = 2c$, it follows that $p$ must divide $c$ as well.

Question 3

Which of the following is a direct application of the Fundamental Theorem of Arithmetic?

Accepted Answer

Correct Option: B. Solution: The Fundamental Theorem of Arithmetic is used to express numbers as products of primes, which can then be used to find the greatest common divisor (GCD) by comparing the smallest powers of common prime factors.

Question 4

If $\sqrt{3}$ were rational, it could be expressed as $\frac{a}{b}$ where $a$ and $b$ are coprime integers. What contradiction arises in the proof of its irrationality?

Accepted Answer

Correct Option: A. Solution: Assuming $\sqrt{3}$ is rational implies $3b^2 = a^2$, leading to the conclusion that both $a$ and $b$ must be divisible by 3, contradicting the assumption that $a$ and $b$ are coprime.

Question 5

If $a$ is a rational number and $b$ is an irrational number, what is the result of the product $a 	imes b$?

Accepted Answer

Correct Option: B. Solution: The product of a non-zero rational number and an irrational number is irrational.

Question 6

In the proof of the irrationality of $\sqrt{2}$, why is it concluded that $a$ and $b$ cannot both be integers with no common factors?

Accepted Answer

Correct Option: B. Solution: In the proof, $2b^2 = a^2$ implies $a$ is divisible by 2, leading to a contradiction since $a$ and $b$ were assumed to be coprime.

Question 7

If a composite number is factorized as $2^3 	imes 3^2 	imes 5 	imes 7 	imes 13$, what is the original number?

Accepted Answer

Correct Option: A. Solution: The number 32760 can be expressed as $2^3 	imes 3^2 	imes 5 	imes 7 	imes 13$, which is its unique prime factorization.

Question 8

According to the theorem, if a prime number $p$ divides $a^2$, what can be concluded about $p$ and $a$?

Accepted Answer

Correct Option: A. Solution: The theorem states that if a prime number $p$ divides $a^2$, then $p$ must also divide $a$.

Question 9

Given that $p$ is a prime number and $p$ divides $a^2$, which of the following statements is true?

Accepted Answer

Correct Option: A. Solution: If $p$ divides $a^2$, then $p$ divides $a$ as per the theorem. Hence, $a$ must be a multiple of $p$.

Question 10

Given the number 32760, which of the following is its correct prime factorization?

Accepted Answer

Correct Option: A. Solution: The prime factorization of 32760 is $2^3 	imes 3^2 	imes 5 	imes 7 	imes 13$, as shown using the factor tree method.

Question 11

Which of the following numbers can be expressed as a product of prime numbers in more than one way?

Accepted Answer

Correct Option: D. Solution: According to the Fundamental Theorem of Arithmetic, the prime factorization of a composite number is unique, apart from the order of the factors. Therefore, none of the numbers can be expressed in more than one way.

Question 12

If the LCM of two numbers is 60 and their HCF is 2, what could be the two numbers?

Accepted Answer

Correct Option: A. Solution: The numbers 6 and 20 have an LCM of 60 and an HCF of 2, as calculated using their prime factors.

Question 13

If $\sqrt{3}$ is assumed to be rational, which equation is derived that leads to a contradiction?

Accepted Answer

Correct Option: A. Solution: Assuming $\sqrt{3}$ is rational leads to $3b^2 = a^2$, implying $a$ is divisible by 3, which contradicts the assumption that $a$ and $b$ are coprime.

Question 14

Using the prime factorization method, determine if 6ⁿ can end with the digit 0 for any natural number n.

Accepted Answer

Correct Option: C. Solution: For a number to end in 0, it must be divisible by 10, which requires a factor of 5. Since the prime factorization of 6ⁿ is $(2 	imes 3)^n$, it contains no factor of 5, hence 6ⁿ cannot end with the digit 0 for any natural number n.

Question 15

Consider the expression $a + b\sqrt{5}$, where $a$ and $b$ are rational numbers and $b 
eq 0$. Which of the following statements is true?

Accepted Answer

Correct Option: B. Solution: The expression $a + b\sqrt{5}$ is irrational because $b\sqrt{5}$ is irrational and adding a rational number $a$ to it cannot result in a rational number, unless $b = 0$, which contradicts the given condition $b 
eq 0$.

Question 16

If the HCF of two numbers is 5 and their product is 200, what is their LCM?

Accepted Answer

Correct Option: B. Solution: Using the relationship HCF(a, b) 	imes LCM(a, b) = a 	imes b, we calculate LCM(a, b) = \frac{200}{5} = 40.

Question 17

Consider a prime number $p$ and a positive integer $a$. If $p$ divides $a^2$, what can be concluded about $p$ and $a$?

Accepted Answer

Correct Option: B. Solution: According to the theorem, if $p$ divides $a^2$, then $p$ must also divide $a$.

Question 18

Which of the following statements correctly describes the Fundamental Theorem of Arithmetic?

Accepted Answer

Correct Option: B. Solution: The Fundamental Theorem of Arithmetic states that every composite number can be expressed as a product of prime numbers, and this factorization is unique apart from the order of the factors.

Question 19

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

Accepted Answer

Correct Option: A. Solution: The Fundamental Theorem of Arithmetic states that every composite number can be expressed uniquely as a product of prime numbers, apart from the order of the factors.

Question 20

Given two numbers, 96 and 404, with HCF as 4, find their LCM.

Accepted Answer

Correct Option: A. Solution: Using the formula HCF(a, b) 	imes LCM(a, b) = a 	imes b, we find LCM(96, 404) = \frac{96 	imes 404}{4} = 9696.

Question 21

Using the prime factorization method, what is the HCF of 6 and 20?

Accepted Answer

Correct Option: B. Solution: The HCF of 6 and 20 is 2, which is the product of the smallest power of each common prime factor.

Question 22

If the HCF of two numbers is 12 and their LCM is 180, what is the product of the two numbers?

Accepted Answer

Correct Option: B. Solution: The relationship between HCF and LCM is given by: $$	ext{HCF}(a, b) 	imes 	ext{LCM}(a, b) = a 	imes b$$. Therefore, the product of the two numbers is $12 	imes 180 = 2160$.

Question 23

Which of the following statements is a misconception regarding the theorem: If $p$ divides $a^2$, then $p$ divides $a$?

Accepted Answer

Correct Option: B. Solution: Option b is a misconception. The theorem explicitly states that if $p$ divides $a^2$, then $p$ must also divide $a$.

Question 24

Find the LCM of 8, 9, and 25 using the prime factorization method.

Accepted Answer

Correct Option: C. Solution: The prime factorization of 8 is $2^3$, 9 is $3^2$, and 25 is $5^2$. The LCM is the product of the highest powers of all prime factors: $2^3 	imes 3^2 	imes 5^2 = 720$.

Question 25

In the proof that $\sqrt{5}$ is irrational, which theorem is applied to show that if $5$ divides $a^2$, then $5$ must divide $a$?

Accepted Answer

Correct Option: C. Solution: Theorem 1.2 states that if a prime $p$ divides $a^2$, then $p$ must divide $a$. This is used in the proof of the irrationality of $\sqrt{5}$.

Question 26

Given the numbers 84 and 120, find the HCF using their prime factorization.

Accepted Answer

Correct Option: A. Solution: The prime factorization of 84 is $2^2 	imes 3 	imes 7$ and for 120 is $2^3 	imes 3 	imes 5$. The HCF is the product of the smallest powers of all common prime factors, which is $2^2 	imes 3 = 12$.

Question 27

Which theorem is crucial for proving the irrationality of $\sqrt{2}$, $\sqrt{3}$, and $\sqrt{5}$?

Accepted Answer

Correct Option: B. Solution: The Fundamental Theorem of Arithmetic is used to prove the irrationality of $\sqrt{2}$, $\sqrt{3}$, and $\sqrt{5}$ by showing that these numbers cannot be expressed as a ratio of two integers.

Question 28

Consider a number $x = 2^4 	imes 3^3 	imes 5$. If the number is multiplied by 7, what will be the prime factorization of the new number?

Accepted Answer

Correct Option: A. Solution: Multiplying the number by 7 adds 7 to the list of prime factors. Therefore, the prime factorization becomes $2^4 	imes 3^3 	imes 5 	imes 7$.

Question 29

Given two numbers 48 and 180, with HCF as 12, find their LCM.

Accepted Answer

Correct Option: A. Solution: Using the formula $$	ext{HCF}(a, b) 	imes 	ext{LCM}(a, b) = a 	imes b$$, we have $12 	imes 	ext{LCM}(48, 180) = 48 	imes 180$. Solving for LCM gives $	ext{LCM}(48, 180) = \frac{48 	imes 180}{12} = 720$.

Question 30

Which statement is true about the Fundamental Theorem of Arithmetic?

Accepted Answer

Correct Option: B. Solution: The Fundamental Theorem of Arithmetic states that every composite number can be expressed as a product of primes uniquely, except for the order.

Question 31

Which of the following numbers cannot be expressed as a product of the primes 2, 3, 5, and 7?

Accepted Answer

Correct Option: B. Solution: The number 231 is not divisible by 5, which is one of the primes listed. Its prime factorization is $3 	imes 7 	imes 11$, which does not include 5.

Question 32

If $x$ is a rational number and $y$ is an irrational number, which of the following operations will always result in an irrational number?

Accepted Answer

Correct Option: A. Solution: The sum of a rational number and an irrational number is always irrational. This is because if the sum were rational, subtracting the rational number from it would yield a rational number, contradicting the irrationality of $y$.

Question 33

Given the prime factorization of two numbers, $a = 2^3 	imes 3^2$ and $b = 2^2 	imes 5$, what is the LCM of $a$ and $b$?

Accepted Answer

Correct Option: A. Solution: The LCM is found by taking the highest power of each prime factor present in the numbers: $2^3 	imes 3^2 	imes 5$.

Question 34

Which theorem is primarily used to prove the irrationality of $\sqrt{2}$, $\sqrt{3}$, and $\sqrt{5}$?

Accepted Answer

Correct Option: B. Solution: The Fundamental Theorem of Arithmetic is used to prove the irrationality of $\sqrt{2}$, $\sqrt{3}$, and $\sqrt{5}$ by showing that these cannot be expressed as a ratio of two integers.

Question 35

If the HCF of two numbers is 4 and their LCM is 60, what is the product of the two numbers?

Accepted Answer

Correct Option: A. Solution: According to the relationship HCF(a, b) 	imes LCM(a, b) = a 	imes b, the product of the two numbers is 4 	imes 60 = 240.

Question 36

Which theorem guarantees that the prime factorization of a composite number is unique?

Accepted Answer

Correct Option: B. Solution: The Fundamental Theorem of Arithmetic states that every composite number can be expressed as a product of primes in a unique way, apart from the order of the factors.

Question 37

If the prime factorization of a number is $2^3 	imes 3^2 	imes 5 	imes 7 	imes 13$, what is the number?

Accepted Answer

Correct Option: A. Solution: The number with the prime factorization $2^3 	imes 3^2 	imes 5 	imes 7 	imes 13$ is 32760.

Question 38

What is the prime factorization of the number 32760?

Accepted Answer

Correct Option: A. Solution: The prime factorization of 32760 is $2^3 	imes 3^2 	imes 5 	imes 7 	imes 13$ as shown in the factor tree.

Question 39

If the LCM of two numbers is 180 and one of the numbers is 36, what is the other number, given that their HCF is 6?

Accepted Answer

Correct Option: B. Solution: Using the relation $	ext{HCF} 	imes 	ext{LCM} = 	ext{Product of the numbers}$, we have $6 	imes 180 = 36 	imes x$. Solving for $x$ gives $x = 60$.

Question 40

If the LCM of two numbers is 252 and their product is 7560, what is their HCF?

Accepted Answer

Correct Option: C. Solution: Using the formula $$	ext{HCF}(a, b) 	imes 	ext{LCM}(a, b) = a 	imes b$$, we have $	ext{HCF}(a, b) 	imes 252 = 7560$. Solving for HCF gives $	ext{HCF}(a, b) = \frac{7560}{252} = 30$.

Question 41

Which of the following expressions is guaranteed to be irrational?

Accepted Answer

Correct Option: A. Solution: The sum of two irrational numbers like $\sqrt{2}$ and $\sqrt{3}$ is generally irrational. In contrast, $\sqrt{2} 	imes \sqrt{3}$ simplifies to $\sqrt{6}$, which is irrational, and $\frac{\sqrt{2}}{\sqrt{3}}$ is also irrational. However, $\sqrt{2} - \sqrt{2}$ simplifies to 0, which is rational.

Question 42

What is the result of adding a rational number to an irrational number?

Accepted Answer

Correct Option: B. Solution: The sum of a rational number and an irrational number is always irrational.

Question 43

If a composite number $x$ is expressed as $2^3 	imes 3^2 	imes 5 	imes 7$, which of the following statements is true about the uniqueness of its prime factorization?

Accepted Answer

Correct Option: B. Solution: According to the Fundamental Theorem of Arithmetic, a composite number can be factorized uniquely as a product of primes, regardless of the order of the primes.

Question 44

Which of the following statements is true regarding the division of a rational number by an irrational number?

Accepted Answer

Correct Option: B. Solution: The division of a non-zero rational number by an irrational number results in an irrational number.

Question 45

According to the Fundamental Theorem of Arithmetic, the prime factorization of a composite number is unique, except for the order of the factors.

Accepted Answer

Answer: True. Solution: The Fundamental Theorem of Arithmetic states that every composite number can be expressed as a product of primes in a unique way, apart from the order in which the prime factors occur.

Question 46

For any two positive integers $a$ and $b$, the product of their HCF and LCM is equal to the product of the numbers themselves.

Accepted Answer

Answer: True. Solution: It is a known result that for any two positive integers $a$ and $b$, $	ext{HCF}(a, b) 	imes 	ext{LCM}(a, b) = a 	imes b$.

Question 47

The theorem stating that if $p$ divides $a^2$, then $p$ divides $a$, is not used in proving the irrationality of numbers like $\sqrt{2}$ or $\sqrt{3}$.

Accepted Answer

Answer: False. Solution: The theorem is crucial in proving the irrationality of numbers like $\sqrt{2}$ and $\sqrt{3}$, as it helps establish that certain assumptions about rationality lead to contradictions.

Question 48

The Fundamental Theorem of Arithmetic states that every composite number can be expressed as a product of prime numbers in a unique way, except for the order of the factors.

Accepted Answer

Answer: True. Solution: The Fundamental Theorem of Arithmetic asserts that every composite number can be factorized uniquely into prime numbers, disregarding the order of these primes.

Question 49

The sum of a rational number and an irrational number is always rational.

Accepted Answer

Answer: False. Solution: The sum of a rational number and an irrational number is always irrational.

Question 50

According to the Fundamental Theorem of Arithmetic, it is possible for a composite number to have more than one unique prime factorization.

Accepted Answer

Answer: False. Solution: The Fundamental Theorem of Arithmetic ensures that the prime factorization of a composite number is unique, apart from the order of the factors.

Question 51

The proof of the irrationality of $\sqrt{2}$ relies on the assumption that it can be expressed as a fraction of two coprime integers.

Accepted Answer

Answer: True. Solution: The proof by contradiction assumes that $\sqrt{2}$ can be expressed as $\frac{a}{b}$, where $a$ and $b$ are coprime integers, and then shows this leads to a contradiction.

Question 52

The HCF of two numbers is obtained by multiplying the greatest power of each prime factor involved in the numbers.

Accepted Answer

Answer: False. Solution: The HCF is obtained by multiplying the smallest power of each common prime factor in the numbers.

Question 53

According to the Fundamental Theorem of Arithmetic, a composite number can have multiple distinct prime factorizations.

Accepted Answer

Answer: False. Solution: The Fundamental Theorem of Arithmetic states that the prime factorization of a composite number is unique, apart from the order of the factors.

Question 54

The Fundamental Theorem of Arithmetic is not used in the proof of the irrationality of $\sqrt{3}$.

Accepted Answer

Answer: False. Solution: The Fundamental Theorem of Arithmetic is used to show that if $3$ divides $a^2$, then it must divide $a$, which is a crucial step in the proof of the irrationality of $\sqrt{3}$.

Question 55

For three positive integers $p$, $q$, and $r$, the product of their HCF and LCM is equal to the product of the integers $p$, $q$, and $r$.

Accepted Answer

Answer: False. Solution: For three numbers, it is not true that $	ext{HCF}(p, q, r) 	imes 	ext{LCM}(p, q, r) = p 	imes q 	imes r$. The relationship holds only for two integers.

Question 56

The product of a non-zero rational number and an irrational number is always an irrational number.

Accepted Answer

Answer: True. Solution: The product of a non-zero rational number and an irrational number is always irrational. This is because multiplying by a non-zero rational number does not eliminate the irrationality.

Question 57

The Fundamental Theorem of Arithmetic is used to prove that $\sqrt{5}$ is irrational.

Accepted Answer

Answer: True. Solution: The Fundamental Theorem of Arithmetic is used in the proof by contradiction to show that if $\sqrt{5}$ were rational, it would lead to a contradiction regarding the uniqueness of prime factorization.

Question 58

The equation $	ext{HCF}(a, b) 	imes 	ext{LCM}(a, b) = a 	imes b$ can be used to find the LCM if the HCF is known.

Accepted Answer

Answer: True. Solution: Given the HCF of two numbers, the LCM can be calculated using the equation $	ext{LCM}(a, b) = \frac{a 	imes b}{	ext{HCF}(a, b)}$.

Question 59

If $p$ is a prime number and $p$ divides $a^2$, then $p$ must divide $a$, where $a$ is a positive integer.

Accepted Answer

Answer: True. Solution: According to Theorem 1.2, if $p$ divides $a^2$, then $p$ divides $a$.

Question 60

The Fundamental Theorem of Arithmetic states that every composite number can be expressed as a product of prime numbers in a unique way, apart from the order of the factors.

Accepted Answer

Answer: True. Solution: The Fundamental Theorem of Arithmetic asserts that every composite number can be expressed as a product of prime numbers uniquely, except for the order of the factors.

Question 61

The product of the HCF and LCM of two positive integers is always equal to the product of the integers themselves.

Accepted Answer

Answer: True. Solution: For any two positive integers $a$ and $b$, the relationship $	ext{HCF}(a, b) 	imes 	ext{LCM}(a, b) = a 	imes b$ holds true.

Question 62

If a prime number $p$ divides $a^2$, then $p$ must also divide $a$.

Accepted Answer

Answer: True. Solution: According to Theorem 1.2, if $p$ divides $a^2$, then $p$ divides $a$. This is based on the Fundamental Theorem of Arithmetic.

Question 63

The proof of the irrationality of $\sqrt{2}$ relies on the assumption that it can be expressed as a fraction of two integers with no common factors.

Accepted Answer

Answer: True. Solution: The proof of the irrationality of $\sqrt{2}$ uses a proof by contradiction, assuming it can be expressed as a fraction of two coprime integers, leading to a contradiction.

Question 64

The Highest Common Factor (HCF) of two numbers is found by taking the product of the greatest power of each prime factor involved in the numbers.

Accepted Answer

Answer: False. Solution: The HCF is found by taking the product of the smallest power of each common prime factor in the numbers.

Real Numbers

Learning Objectives

Learning Objectives

Revision Notes & Summary

Chapter Notes

Euclid's Division Algorithm

Fundamental Theorem of Arithmetic

Prime Factorization Method for HCF and LCM

Proof of Irrationality Using Fundamental Theorem

Properties of Rational and Irrational Numbers

Exam Tips & Common Mistakes

Common Mistakes & Exam Tips

Euclid's Division Algorithm

Fundamental Theorem of Arithmetic

Prime Factorization Method for HCF and LCM

Proof of Irrationality Using Fundamental Theorem

Properties of Rational and Irrational Numbers

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Multiple Choice Questions

Q1.If ppp is a prime number and ppp divides a2a^2a2, which of the following is a valid application of this theorem?

Correct Answer: A

Q2.Assume ppp is a prime number and ppp divides a2a^2a2. If a=2ca = 2ca=2c for some integer ccc, what can be inferred about ccc?

Correct Answer: A

Q3.Which of the following is a direct application of the Fundamental Theorem of Arithmetic?

Correct Answer: B

Q4.If 3\sqrt{3}3​ were rational, it could be expressed as ab\frac{a}{b}ba​ where aaa and bbb are coprime integers. What contradiction arises in the proof of its irrationality?

Correct Answer: A

Q5.If aaa is a rational number and bbb is an irrational number, what is the result of the product a×ba \times ba×b?

Correct Answer: B

Q6.In the proof of the irrationality of 2\sqrt{2}2​, why is it concluded that aaa and bbb cannot both be integers with no common factors?

Correct Answer: B

Q7.If a composite number is factorized as 23×32×5×7×132^3 \times 3^2 \times 5 \times 7 \times 1323×32×5×7×13, what is the original number?

Correct Answer: A

Q8.According to the theorem, if a prime number ppp divides a2a^2a2, what can be concluded about ppp and aaa?

Correct Answer: A

Q9.Given that ppp is a prime number and ppp divides a2a^2a2, which of the following statements is true?

Correct Answer: A

Q10.Given the number 32760, which of the following is its correct prime factorization?

Correct Answer: A

Q11.Which of the following numbers can be expressed as a product of prime numbers in more than one way?

Correct Answer: D

Q12.If the LCM of two numbers is 60 and their HCF is 2, what could be the two numbers?

Correct Answer: A

Q13.If 3\sqrt{3}3​ is assumed to be rational, which equation is derived that leads to a contradiction?

Correct Answer: A

Q14.Using the prime factorization method, determine if 6ⁿ can end with the digit 0 for any natural number n.

Correct Answer: C

Q15.Consider the expression a+b5a + b\sqrt{5}a+b5​, where aaa and bbb are rational numbers and b≠0b \neq 0b=0. Which of the following statements is true?

Correct Answer: B

Q16.If the HCF of two numbers is 5 and their product is 200, what is their LCM?

Correct Answer: B

Q17.Consider a prime number ppp and a positive integer aaa. If ppp divides a2a^2a2, what can be concluded about ppp and aaa?

Correct Answer: B

Q18.Which of the following statements correctly describes the Fundamental Theorem of Arithmetic?

Correct Answer: B

Q19.According to the Fundamental Theorem of Arithmetic, which of the following statements is true?

Correct Answer: A

Q20.Given two numbers, 96 and 404, with HCF as 4, find their LCM.

Correct Answer: A

Q21.Using the prime factorization method, what is the HCF of 6 and 20?

Correct Answer: B

Q22.If the HCF of two numbers is 12 and their LCM is 180, what is the product of the two numbers?

Correct Answer: B

Q23.Which of the following statements is a misconception regarding the theorem: If ppp divides a2a^2a2, then ppp divides aaa?

Correct Answer: B

Q24.Find the LCM of 8, 9, and 25 using the prime factorization method.

Correct Answer: C

Q25.In the proof that 5\sqrt{5}5​ is irrational, which theorem is applied to show that if 555 divides a2a^2a2, then 555 must divide aaa?

Correct Answer: C

Q26.Given the numbers 84 and 120, find the HCF using their prime factorization.

Correct Answer: A

Q27.Which theorem is crucial for proving the irrationality of 2\sqrt{2}2​, 3\sqrt{3}3​, and 5\sqrt{5}5​?

Correct Answer: B

Q28.Consider a number x=24×33×5x = 2^4 \times 3^3 \times 5x=24×33×5. If the number is multiplied by 7, what will be the prime factorization of the new number?