Class 10th · Mathematics · Chapter 1

Real Numbers – Notes, MCQs, Quiz & Worksheet

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Overview

What is Real Numbers?

Real numbers are the set of numbers that include both rational and irrational numbers. They can be plotted on a number line, which makes them easy to understand and visualize.

Types of Real Numbers:

  • Natural Numbers (N): Numbers we use for counting. Examples: 1, 2, 3, 4, …
  • Whole Numbers (W): Natural numbers + 0. Examples: 0, 1, 2, 3, …
  • Integers (Z): Whole numbers and their negatives. Examples: … -3, -2, -1, 0, 1, 2, 3 …
  • Rational Numbers (Q): Numbers that can be written as a fraction of two integers (denominator ≠ 0).
    Examples: ½, -3/4, 0.25 (which is 1/4), 5 (which is 5/1)
  • Irrational Numbers: Numbers that cannot be expressed as a simple fraction. Their decimal expansions are non-terminating and non-repeating. Examples: √2, π (pi), e

Properties of Real Numbers:

  • Commutative Property:
    • a + b = b + a
    • a × b = b × a
  • Associative Property:
    • (a + b) + c = a + (b + c)
    • (a × b) × c = a × (b × c)
  • Distributive Property:
    • a × (b + c) = a × b + a × c
  • Identity Elements:
    • 0 is the additive identity
    • 1 is the multiplicative identity
  • Closure Property:
    • The result of adding, subtracting, multiplying, or dividing (except by 0) two real numbers is always a real number.

Exam relevance

Real Numbers carries steady weightage in Class 10th exams. Practising its MCQs and important questions is one of the fastest ways to secure marks from this chapter.

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MCQ Practice

Practice MCQs – Real Numbers

Attempt these multiple-choice questions, then reveal the answer to check yourself.

Q1.In a school there are two sections – section A and section B of class X. There are 32 students in section A and 36 students in section B. Determine the minimum number of books required for their class library so that they can be distributed equally among students of section A or section B.

To find the minimum number of books that can be equally distributed among:

32 students in section A
36 students in section B
This means we are looking for the Least Common Multiple (LCM) of 32 and 36.

Step 1: Prime Factorization

32 = 2⁵
36 = 2² × 3²

Step 2: Find LCM

LCM takes the highest powers of all prime factors involved:
LCM = 2⁵ × 3² = 32 × 9 = 288

Hence answer is 288 books

Q2.Find the smallest number that when divided by 12, 16, and 24 leaves a remainder of 4 in each case.
A.52
B.45
C.54
D.44
Answer: 52
Q3.Find the least number which when divided by 6, 15 and 18 leave remainder 5 in each case.

To solve this, we need to find the least number which, when divided by 6, 15, and 18, leaves a remainder of 5 in each case.

Step 1:  Find LCM of 6, 15, and 18

  • 6     =   2  ×  3
  • 15   =  3   ×  5
  • 18   =  2   ×   32

Take the highest powers of all primes:
LCM     =21  ×  32  ×   51   =   90

So, the least number is: 90 + 5 = 95

Q4.The product of HCF and LCM of 60,84 and 108 is
A.55360
B.55390
C.78902
D.45360
Answer: 45360
Q5.Determine the number nearest 110000 but greater than 100000 which is exactly divisible by each of 8, 15 and 21.

Step 1: Find the Least Common Multiple (LCM) of 8, 15, and 21.

Prime factorization:
8 = 2³
15 = 3 × 5
21 = 3 × 7
LCM = 2³ × 3 × 5 × 7 = 8 × 3 × 5 × 7 = 840

So, any number divisible by 8, 15, and 21 must be a multiple of 840.

Step 2: Find the multiple of 840 that is nearest to 110000 and greater than 100000.

Divide 110000 by 840:
110000 ÷ 840 ≈ 130.95
Now check nearby multiples:
130 × 840 = 109200
131 × 840 = 110040
132 × 840 = 110880

Among these, 110040 is the nearest to 110000 and is greater than 100000.

Hence answer is 110040

Q6.Three tankers contain 403 litres, 434 litres and 465 litres of diesel respectively. Find the maximum capacity of a container that can measure the diesel of the three containers exact number of times.

To solve this problem, we need to find the greatest common divisor (GCD) of the three given quantities: 403 litres, 434 litres, and 465 litres.

The GCD will be the maximum capacity of a container that can measure all three amounts an exact number of times.

Find the GCD of 403, 434, and 465

Step 1:   GCD of 403 and 434
                 Use the Euclidean algorithm:
                 434 ÷ 403 = 1 remainder 31
                 403 ÷ 31 = 13 remainder 0

                 So, GCD(403, 434) = 31

Step2:   Now, GCD 31 and 465
                465 ÷ 31 = 15 remainder 0

                 So, GCD(31, 465) = 31

Hence the maximum capacity of the container that can measure all three diesel quantities exactly is 31 litres.

Q7.If the HCF of two numbers is 16 and their product is 3072, find their LCM.
A.192
B.182
C.172
D.162
Answer: 192
Q8.Find the smallest number that is divisible by 6, 10, and 15, and leaves a remainder of 5 in each case.
A.35
B.25
C.45
D.55
Answer: 35
Q9.If HCF(6, a) = 2 and LCM(6, a) = 60 then find the value of a.

It is given -

  • HCF of 6 & a is 2
  • LCM of 6 & a is 60

Now we use the identity 

HCF x LCM = Product of both Number

2 x 60 = 6 x a

=> a = 20

Final Answer: 20

 

Q10.The LCM of 60, 84 and 108 is
A.3780
B.4789
C.2300
D.3790
Answer: 3780
Q11.Prove that 7×11×13 +13 is a composite number.

7× 11 × 13 + 13 can be wriiten as 13 (7× 11 + 1).

=> 13 x (77 + 1)

=> 13 x 78

=> 13 x 2 x 39

Clearly, this has more than 2 factors.

So, 7× 11 × 13 + 13 is a composite number.

Q12.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
A.2
B.6
C.4
D.12
Answer: 4
Q13.Which of the following is a rational number?
A.square root of 3
B.pi
C.0.1010010001...
D.22/7
Answer: 22/7
Q14.According to Euclid's Division Lemma, for two positive integers 'a' and 'b', there exist unique integers 'q' and 'r' such that a = bq + r. What is the condition for 'r'?
A.0 < r < b
B.0 <= r <= b
C.0 <= r < b
D.0 < r <= b
Answer: 0 <= r < b
Q15.The product of a non-zero rational number and an irrational number is always:
A.Rational
B.Irrational
C.Rational or Irrational
D.Always an integer
Answer: Irrational
Q16.The decimal expansion of the rational number 33/2*2 * 5 will terminate after:
A.One decimal place
B.Two decimal places
C.Three decimal places
D.Four decimal places
Answer: Two decimal places
Q17.If HCF(26, 91) = 13, then LCM(26, 91) is:
A.13
B.182
C.2366
D.91
Answer: 182
Q18.Every composite number can be expressed as a product of primes, and this factorization is unique, apart from the order in which the prime factors occur. This statement is known as:
A.Euclid's Division Lemma
B.Fundamental Theorem of Arithmetic
C.Rational Number Theorem
D.Irrational Number Theorem
Answer: Fundamental Theorem of Arithmetic
Q19.What is the HCF of the smallest prime number and the smallest composite number?
A.1
B.2
C.3
D.4
Answer: 2
Q20.Which of the following is an irrational number?
A.square root of 4
B.0.333...
C.7/8
D.3 + square root of 2
Answer: 3 + square root of 2
Q21.If 'p' is a prime number, then square root of p is:
A.Rational
B.Irrational
C.An integer
D.A natural number
Answer: Irrational
Q22.The decimal expansion of the number 147/1200 will be:
A.Terminating
B.Non-terminating repeating
C.Non-terminating non-repeating
D.Cannot be determined
Answer: Terminating
Q23.How many prime factors are there in the prime factorization of 200?
A.2
B.3
C.4
D.5
Answer: 2
Q24.For any positive integer 'n', n(n+1) is always:
A.Odd
B.Even
C.Prime
D.Composite
Answer: Even
Q25.If two positive integers 'a' and 'b' are written as a = x³*y² and b = x*y³, where x, y are prime numbers, then HCF(a, b) is:
A.x*y
B.x*y²
C.x³*y³
D.x²*y²
Answer: x*y²
Q26.If two positive integers 'a' and 'b' are written as a = x³*y² and b = x*y³, where x, y are prime numbers, then LCM(a, b) is:
A.x*y
B.x*y²
C.x³*y³
D.x²*y²
Answer: x³*y³
Q27.The sum of a rational and an irrational number is:
A.Always rational
B.Always irrational
C.Rational or irrational
D.Always an integer
Answer: Always irrational
Q28.The square of any positive integer cannot be of the form:
A.3m
B.3m + 1
C.5m + 2
D.5m
Answer: 5m + 2
Q29.Which of the following is not a rational number?
A.0.14
B.0.14 bar (repeating)
C.0.141141114...
D.0.141414...
Answer: 0.141141114...
Q30.If 'a' and 'b' are coprime, then HCF(a, b) is:
A.a
B.b
C.ab
D.1
Answer: 1
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Important Questions

Most Expected Real Numbers Questions

High-weightage and frequently asked questions to prioritise before exams.

Quick Revision

Real Numbers – Quick Revision Notes

A one-page recap to revise the whole chapter in minutes.

  • Real Numbers is part of the Class 10th Mathematics syllabus and carries steady exam weightage.
  • Re-read all formulas, laws and definitions from this chapter.
  • Re-attempt the MCQs you got wrong and solve one worksheet.
  • Finish with a short quiz to confirm you remember everything.
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