Class X – Mathematics Chapter 1: Real Numbers


Class X – Mathematics
Chapter 1: Real Numbers

In this chapter, questions and answers are designed to test both conceptual understanding and problem-solving skills. The questions are mainly based on logical steps, proper use of theorems, and clear mathematical reasoning.

Most questions come from the following areas:

  • Euclid’s Division Lemma and Algorithm
    Questions involve finding the HCF of given numbers and solving simple word problems using step-by-step division.

  • Fundamental Theorem of Arithmetic
    You are asked to express numbers as products of prime factors and use this idea to find HCF and LCM.

  • HCF and LCM Relationship
    Some questions test the relation
    HCF × LCM = Product of two numbers.

  • Decimal Expansion of Rational Numbers
    These questions check whether a given rational number has a terminating or non-terminating recurring decimal expansion and ask for proper justification.

  • Proof-based Questions
    A few questions require short proofs, such as showing why a given number is composite or why a certain form of number is not possible.

How to Write Answers in Exams

  • Write steps clearly and in order
  • Mention theorem or lemma names where required
  • Show proper calculations and reasoning
  • Avoid skipping steps, even if the question looks easy

Overall, the questions and answers in this chapter help build a strong base in number theory and improve logical thinking, which is essential for higher classes.

Question 1

Use Euclid’s Division Algorithm to find the HCF of 135 and 225.

Solution

225 ÷ 135 = 1 remainder 90
135 ÷ 90 = 1 remainder 45
90 ÷ 45 = 2 remainder 0

Since the remainder is 0,
HCF = 45

Question 2

Find the HCF of 306 and 657 using Euclid’s Division Algorithm.

Solution

657 ÷ 306 = 2 remainder 45
306 ÷ 45 = 6 remainder 36
45 ÷ 36 = 1 remainder 9
36 ÷ 9 = 4 remainder 0

HCF = 9

Question 3

Show that any positive odd integer is of the form 6q + 1, 6q + 3, or 6q + 5.

Solution

By Euclid’s Division Lemma, any integer a can be written as:
a = 6q + r, where r = 0,1,2,3,4,5

Odd integers cannot be divisible by 2, so r ≠ 0,2,4

Thus possible values of r are 1, 3, 5

Hence, any positive odd integer is of the form:
6q + 1, 6q + 3, or 6q + 5

Question 4

Find the LCM and HCF of 306 and 657 using prime factorization.

Solution

306 = 2 × 3² × 17
657 = 3² × 73

HCF = 3² = 9
LCM = 2 × 3² × 17 × 73 = 37962

Question 5

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

Solution

Prime factorization of 6ⁿ = (2 × 3)ⁿ = 2ⁿ × 3ⁿ

For a number to end in 0, it must have both 2 and 5 as factors.
Here, factor 5 is missing.

So, 6ⁿ cannot end with 0 for any natural number n.

Question 6

Express 140 as a product of its prime factors.

Solution

140 = 14 × 10
= (2 × 7) × (2 × 5)
= 2² × 5 × 7

Question 7

Find the HCF and LCM of 20 and 28 and verify that
HCF × LCM = Product of the numbers.

Solution

20 = 2² × 5
28 = 2² × 7

HCF = 2² = 4
LCM = 2² × 5 × 7 = 140

HCF × LCM = 4 × 140 = 560
Product of numbers = 20 × 28 = 560

✔ Verified

Question 8

Find whether the decimal expansion of 13/125 is terminating or non-terminating.

Solution

125 = 5³

The denominator has only factor 5.
So the decimal expansion is terminating.

Question 9

Find whether the decimal expansion of 17/90 is terminating or non-terminating.

Solution

90 = 2 × 3² × 5

Since the denominator contains 3,
the decimal expansion is non-terminating recurring.

Question 10

Explain why 7 × 11 × 13 + 13 is a composite number.

Solution

7 × 11 × 13 + 13
= 13(7 × 11 + 1)
= 13 × 78

Since it has factors other than 1 and itself,
the number is composite.

Exam Tip

✔ These questions cover all important concepts
✔ Practice steps clearly for full marks
✔ Statements + reasoning matter a lot in this chapter


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