01 Real Numbers Class-X ( AP / TG / CBSE )

1.1.Introduction

1.2.The Fundamental Theorem of Arithmetic  

Theorem 1.1 (Fundamental Theorem of Arithmetic) : Every composite number can be expressed (factorised) as a product of primes, and this factorisation is unique, apart from the order in which the prime factors occur.

Applications of FTA

1. To determine the unit's or one's digit of a number given in exponential form.

2.To find LCM/HCF  of two or more numbers.

 HCF = Product of the smallest power of each common prime factor in the numbers ( when written in Prime Factorization form )

LCM = Product of the greatest power of each prime factor, involved in the numbers ( when written in Prime Factorization form )

Example 1 : Consider the numbers 4n , where n is a natural number. Check whether there is any value of n for which 4ⁿ ends with the digit zero.

Example 2 : Find the LCM and HCF of 6 and 20 by the prime factorisation method.   [AP SQP-2025 ]

Example 3: Find the HCF of 96 and 404 by the prime factorisation method. Hence, find their LCM.

Example 4 : Find the HCF and LCM of 6, 72 and 120, using the prime factorisation method.

Note: The product of three numbers is not equal to the product of their HCF and LCM.

EXERCISE 1.1 

1.Express each number as a product of its prime factors: 

    (i)140 
    (ii)156    [AP SQP 2025 ]
    (iii)3825 
    (iv)5005
    (v) 7429

2. Find the LCM and HCF f the following pairs of integers and verify that LCM x HCF = product of the two numbers.

     (i) 26 and 91

     (ii) 510 and 91 

     (iii) 336 and 54

 

3.Find the LCM and HCF of the following integers by applying the prime factorisation method

     (i) 12,15 and 21

     (ii) 17,23 and 29

     (iii) 8,9 and 25

4. Given that HCF (306, 657) = 9, find LCM (306, 657).  [AP HAP ]

5. Check whether 6ⁿ  can end with the digit 0 for any natural number n.

6. Explain why 7 × 11 × 13 + 13 and 7 × 6 × 5 × 4 × 3 × 2 × 1 + 5 are composite numbers. [AP HAP]

7. There is a circular path around a sports field. Sonia takes 18 minutes to drive one round of the field, while Ravi takes 12 minutes for the same. Suppose they both start at the same point and at the same time, and go in the same direction. After how many minutes will they meet again at the starting point?

More Practice-1.1

1.Express each number as a product of its prime factors( Prime Factorization ) : 

     (i) 253 
    (ii) 1771 
    (iii) 5313 
    (iv) 10626  
    (v)  8232 
    (vi) 21252 
    (vii) 32760  
    (viii ) 30  [AP SSC SQP 2025 ]

1. What is the HCF of 37 and 49   [AP SSC March 2024]
2. If HCF (26,91) is 13, then find LCM of (26,91)   [AP SSC March 2025]
3. State the Fundamental Theorem of Arithmetic. [AP HAP]
4. Find the smallest pair of 4-digit numbers such that the difference between them is 303 and their HCF is 101. Show your steps 
5. Find the H.C.F and L.C.M of 480 and 720 using the Prime factorisation method. [CBSE SQP ]
6. The H.C.F of 85 and 238 is expressible in the form 85m -238. Find the value of m. [CBSE SQP ]

Objective Questions-1.1

1.If two positive integers a and b are written as a = x³y²  and b = xy³, where x, y are prime numbers, then the result obtained by dividing the product of the positive integers by the LCM (a, b) is    

1. xy
2. xy²  
   
3. x³y³
4. x²y²  

2.The LCM of smallest two digit composite number and smallest composite number is:  

1. 12
2. 4 
   
3. 20 
4. 44

3.If two positive integers a and b are written as a = x³y² and b = xy³; where x, y are prime numbers, then HCF (a,b) is:    

1. xy
2. xy²
3.  x³y³
4. x²y

4.If n is a natural number (n ∈ N) then the Units place digit of 6ⁿ   for natural value of n is ______
5.Full form of HCF is _______

6. Which of these numbers can be expressed as a product of two or more prime numbers?
    i) 15
    ii) 34568
    iii) (15 × 13)

1. only (ii)
2.  only (iii)
3.  only (i) and (ii)
4.  all - (i), (ii) and (iii)

7.1245 is a factor of the numbers p and q. Which of the following will ALWAYS have 1245 as a factor?
    (i) p + q
    (ii) p × q
    (iii) p ÷ q

1. only (ii)
2. only (i) and (ii)
3. only (ii) and (iii)
4. all - (i), (ii) and (iii)

8.Which of the following numbers can be written as a non-terminating but recurring decimal?

1. 9
2.  43/8
3.  √6
4.  5/12

9.HCF of (3³×5² × 2), (3²   × 5³  × 2²   ) 𝑎𝑛𝑑 (3⁴ × 5 × 2³  ) is   [CBSE SQP]

     (A) 450 (B) 90 (C) 180 (D) 630

10.If H.C.F(420,189) = 21 then L.C.M(420,189) is   [CBSE SQP ]

     (A) 420 (B) 1890 (C) 3780 (D) 3680

11.Assertion (A): HCF of any two consecutive even natural numbers is always 2. 

    Reason (R): Even natural numbers are divisible by 2.     [CBSE SQP ]

    Choose the correct option   

    A)    Both assertion (A) and reason (R) are true and reason (R) is the correct explanation of assertion     (A)     B)    Both assertion (A) and reason (R) are true and reason (R) is not the correct explanation of     assertion     (A)    C)Assertion (A) is true but reason (R) is false.    D)Assertion (A) is false but reason     (R) is true. 

12. Assertion (A): The HCF of two numbers is 5 and their product is 150. Then their LCM is 40.
Reason(R): For any two positive integers a and b, HCF (a, b) x LCM (a, b) = a x b.   

1. Both assertion (A) and reason (R) are true and reason (R) is the correct explanation of assertion (A).
2. Both assertion (A) and reason (R) are true but reason (R) is not the correct explanation of assertion (A).
3. Assertions (A) is true but reason (R) is false.
4. Assertions (A) is false but reason (R) is true.

13. P and Q are two positive integers such that P = 𝑝³𝑞 and Q = (𝑝𝑞)², where p and q are prime numbers. What is LCM(P, Q)?

1. 𝑝𝑞
2. 𝑝²𝑞²
3.  𝑝³𝑞²
4.  𝑝⁵𝑞³

1.3.Revising Irrational Numbers

Theorem 1.2 : Let p be a prime number. If p divides a² , then p divides a, where a is a positive integer

Theorem 1.3 : √2  is irrational. 

Example 5 : Prove that √3  is irrational    [CBSE SQP ] [AP-HAP, AP SQP 2025 ]

Example 6 : Show that 5– √3  is irrational. 

Example 7 : Show that 3 √2  is irrational

EXERCISE 1.2 

1.Prove that  √5 is irrational    [AP SQP 2025] [CBSE SQP ]

2.Prove that 3+2√5 is irrational   [AP HAP(8)]

3.Prove that the following are irrational

  (i) 1/√2    

  (ii) 7√5   

  (iii) 6+√2

 More Practice-1.2

1.Prove that √7 is irrational number. [AP SQP 2025, AP SSC MARCH 2024, 2025 ]

2.Prove that 2+√3 is irrational [AP SSC SQP 2025 ]

Objective Questions- 1.2 

1.Statement A : 22/7  =  𝜋  (approximately) is irrational.     [ AP HAP]

 Statement B : All Non-terminating and Non-repeating decimals are irrational.
(A) Both Statements A and B are correct
(B) Only statement A is True
(C) Only statement B is True
( D ) Both statements A, B are false. 

2. Statement A :𝜋  is irrational.    [AP SQP 2025 ]

Statement B : All non-terminating and non-repeating decimals are irrational.
(A) Both statement A and B are true
(B) Only statement A is true
(C) Only statement B is true
(D) Both statements A and B are false

3.Which of these is a RATIONAL number?

1. 3π
2. 5√5
3. 0.3466666...
4.  0.345210651372849...

4.  The fraction 7/𝑞 has a terminating decimal expansion. Which of these CANNOT be q?

1. 8 × 2
2.  8 × 3
3.  8 × 4
4.  8 × 5

5. Which of the following will have the MAXIMUM number of 6's when written in decimal form?

1. 666/1000
2. 3/6
3. 3/5
4. 2/3

6.Two statements are given below - one labelled Assertion (A) and the other labelled Reason (R). Read the statements carefully and choose the option that correctly describes statements (A) and (R).
    Assertion (A): 2 is a prime number.
    Reason (R): The square of an irrational number is always a prime number.

1.  Both (A) and (R) are true and (R) is the correct explanation of (A).

1.  Both (A) and (R) are true and (R) is not the correct explanation of (A).

1.  (A) is true but (R) is false.

1.  (A) is false but (R) is true.

7. DIRECTION: In this question  a statement of Assertion (A) is followed by a statement of Reason (R). Choose the correct option 
A) Both assertion (A) and reason (R) are true and reason (R) is the correct explanation of assertion (A) 
B) Both assertion (A) and reason (R) are true and reason (R) is not the correct explanation of assertion (A) 
C) Assertion (A) is true but reason (R) is false. 
D) Assertion (A) is false but reason (R) is true.   [CBSE SQP ]
Assertion(A): (2 + √3)√3 is an irrational number. 
Reason(R): Product of two irrational numbers is always irrational.

1.4. Summary

1. The Fundamental Theorem of Arithmetic : Every composite number can be expressed (factorised) as a product of primes, and this factorisation is unique, apart from the order in which the prime factors occur. 

2. If p is a prime and p divides a², then p divides a, where a is a positive integer. 

3. To prove that  √2, √3  are irrationals.

Case Study Based Questions 

Case Study-1

A Mathematics Exhibition is being conducted in your School and one of your friends is making a model of a factor tree. He has some difficulty and asks for your help in completing a quiz for the audience.
Observe the following factor tree and answer the following:

 1 What will be the value of x?

1. 15005 
2. 13915   
3. 56920   
4. 17429

2. What will be the value of y?

1. 23
2. 22
3. 11 
4. 19

3. What will be the value of z?

1. 22    
2. 23   
3.  17   
4.  19

4. According to Fundamental Theorem of Arithmetic 13915 is a........    

1. Composite number
2. Prime number
3. Neither prime nor composite
4. Even number 

5.The prime factorization of 13915 is ..........   

1. a) 5x11³x13² 
2. 5x11³x23²
3. 5x11²x23
4. 5 x 11² x 13²

Case Study-2

     To enhance the reading skills of grade X students, the school nominates you and two of your friends to set up a class library. There are two sections- section A and section B of grade X. There are 32 students in section A and 36 students in section B. 

1.What is the minimum number of books you will acquire for the class library, so that they can be distributed equally among students of Section A or Section B?     

1. 144
2. 128
3. 288
4. 272 

2. If the product of two positive integers is equal to the product of their HCF and LCM is true then, the HCF (32 , 36) is ........... 

1.  2   
2. 4
3. 6
4. 8 

3. 36 can be expressed as a product of its primes as

1. 2²x3²
2. 2¹x3³
3. 2³x3¹
4. 2⁰x3⁰

4. 7 is a    

1. Prime number
2. Composite number
3. Neither prime nor composite  
4. None of the above 

5. If p and q are positive integers such that p = ab² and q= a²b, where a , b are prime numbers, then the LCM (p, q) is a)

1. ab
2. a²b²  
3. a³b²   
4. a³b³
   

 Case Study-3

     A seminar is being conducted by an Educational Organization, where the participants will be educators of different subjects. The number of participants in Hindi, English and Mathematics are 60, 84 and 108 respectively.

1. In each room the same number of participants are to be seated and all of them being in the same subject, hence maximum number participants that can accommodated in each room are  

1. 14
2.   12
3. 16
4. 18 

2. What is the minimum number of rooms required during the event? 

1. 11    
   
2.  31
3. 41
4. 21 

3. The LCM of 60, 84 and 108 is 

1. 3780
2.  3680
3.   4780
4.  4680 

4. The product of HCF and LCM of 60,84 and 108 is 

1. 55360
2. 35360
3. 45500
4. 45360 

5. 108 can be expressed as a product of its primes as  

1. 2²x3²
2. 2³x3³ 
3. 2³x3² 
4. 2² x3³