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About Real Numbers — Class 10 CBSE

Apply Euclid's division lemma and fundamental theorem of arithmetic to HCF, LCM, and irrationality proofs. This topic is part of the CBSE Class 10 mathematics syllabus (chapter: Chapter 1). On this page you can practice 60 questions across three difficulty levels — 20 easy, 20 medium, and 20 hard — each with a visual step-by-step solution, plus a timed 35-question mastery test with a live leaderboard. Worked examples and sample questions from the topic are below.

What you'll learn in Real Numbers

  • Euclid's Division Lemma
  • The Fundamental Theorem of Arithmetic
  • Revisiting Irrational Numbers
  • Decimal Expansions of Rational Numbers
  • Operations with Irrational Numbers

Interactive lesson · about 20 minutes · checkpoint question after every unit

Real Numbers — solved examples for Class 10 CBSE

Example 1easy

What is the HCF of 12 and 18?
  1. A)2
  2. B)4
  3. C)6
  4. D)12

Step-by-step solution

  1. Find the factors of each number:
  2. Factors of 12:
    12=2x2x312 = 2 x 2 x 3
  3. Factors of 18:
    18=2x3x318 = 2 x 3 x 3
  4. Common factors: 2 and 3
    HCF=2x3=6HCF = 2 x 3 = 6

Answer: 6

Example 2medium

Find the HCF of 867 and 255 using Euclid's division algorithm.
  1. A)51
  2. B)17
  3. C)3
  4. D)85

Step-by-step solution

  1. Step 1:
    867=255x3+102867 = 255 x 3 + 102
  2. Step 2:
    255=102x2+51255 = 102 x 2 + 51
  3. Step 3:
    102=51x2+0102 = 51 x 2 + 0
  4. Remainder is 0, so HCF = 51

Answer: 51

Example 3hard

Prove that sqrt(2) + sqrt(3) is irrational. If we assume it's rational (= a/b), what do we get when we square both sides?
  1. A)a^2/b^2 = 5 + 2sqrt(6)
  2. B)a^2/b^2 = 5 + sqrt(6)
  3. C)a^2/b^2 = 6
  4. D)a^2/b^2 = 2 + 3

Step-by-step solution

  1. Assume sqrt(2) + sqrt(3) = a/b (rational)
  2. Square both sides:
    (sqrt(2)+sqrt(3))2=a2/b2(sqrt(2) + sqrt(3))^2 = a^2/b^2
  3. Expand:
    2+2sqrt(6)+3=a2/b22 + 2sqrt(6) + 3 = a^2/b^2
  4. So:
    5+2sqrt(6)=a2/b25 + 2sqrt(6) = a^2/b^2
  5. This gives sqrt(6) = (a^2/b^2 - 5)/2, making sqrt(6) rational -- contradiction!

Answer: a^2/b^2 = 5 + 2sqrt(6)

Practice questions on Real Numbers

  1. Q1.easy

    The LCM of 4 and 6 is:
    1. A)6
    2. B)12
    3. C)24
    4. D)8
    Show answer

    Answer: 12

    Hint: LCM is the smallest number divisible by both 4 and 6.

  2. Q2.easy

    Which of the following is an irrational number?
    1. A)3/7
    2. B)0.25
    3. C)sqrt(5)
    4. D)0.333...
    Show answer

    Answer: sqrt(5)

    Hint: An irrational number cannot be expressed as p/q where p, q are integers.

  3. Q3.easy

    Using Euclid's division algorithm, the HCF of 455 and 42 starts with which division?
    1. A)455 = 42 x 10 + 35
    2. B)42 = 455 x 0 + 42
    3. C)455 = 42 x 11 + 3
    4. D)455 = 42 x 10 + 25
    Show answer

    Answer: 455 = 42 x 10 + 35

    Hint: Divide the larger number by the smaller and find the quotient and remainder.

  4. Q4.medium

    Find the LCM of 12, 15, and 21.
    1. A)420
    2. B)180
    3. C)360
    4. D)840
    Show answer

    Answer: 420

    Hint: Find prime factorisations of all three and take highest powers.

  5. Q5.medium

    Prove that sqrt(3) is irrational. Which assumption starts the proof?
    1. A)Assume sqrt(3) is an integer
    2. B)Assume sqrt(3) = p/q where p, q are co-prime integers
    3. C)Assume sqrt(3) is negative
    4. D)Assume sqrt(3) = 0
    Show answer

    Answer: Assume sqrt(3) = p/q where p, q are co-prime integers

    Hint: This proof uses contradiction -- start by assuming the opposite of what you want to prove.

  6. Q6.medium

    The HCF of 65 and 117 is expressible in the form 65m - 117. Find the value of m.
    1. A)1
    2. B)2
    3. C)3
    4. D)4
    Show answer

    Answer: 2

    Hint: First find HCF(65, 117), then solve 65m - 117 = HCF.

  7. Q7.hard

    Find the largest number that divides 2053 and 967 and leaves remainders 5 and 7 respectively.
    1. A)120
    2. B)128
    3. C)64
    4. D)32
    Show answer

    Answer: 64

    Hint: The number divides (2053-5) and (967-7) exactly. Find HCF of those results.

  8. Q8.hard

    Use Euclid's division lemma to show that the square of any positive integer is of the form 3m or 3m+1.
    1. A)By dividing by 2
    2. B)By expressing integer as 3q, 3q+1, or 3q+2 and squaring each
    3. C)By using induction
    4. D)By prime factorisation
    Show answer

    Answer: By expressing integer as 3q, 3q+1, or 3q+2 and squaring each

    Hint: Any integer can be written as 3q, 3q+1, or 3q+2 by Euclid's lemma with b=3.

  9. Q9.hard

    Show that the cube of any positive integer is of the form 9m, 9m+1, or 9m+8. How many cases need to be checked?
    1. A)2
    2. B)3
    3. C)4
    4. D)9
    Show answer

    Answer: 3

    Hint: Use Euclid's lemma with b=3 to get three possible forms for the integer.

These are 9 of the 60 questions available for Real Numbers. Start practicing above to unlock visual solutions, AI coaching, and the topic leaderboard — completely free.