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Mathematics · Textbook solutions

Real Numbers

Every solved example, exercise, and miscellaneous question — in the order the textbook teaches them. · 27 questions

1.2 The Fundamental Theorem of Arithmetic

19 q

Solved Examples

Worked · 4
  1. 1.1 Eg.1
    Consider the numbers 4n4^n, where nn is a natural number. Check whether there is any value of nn for which 4n4^n ends with the digit zero.
  2. 1.1 Eg.2
    Find the LCM and HCF of 6 and 20 by the prime factorisation method.
  3. 1.1 Eg.3
    Find the HCF of 96 and 404 by the prime factorisation method. Hence, find their LCM.
  4. 1.1 Eg.4
    Find the HCF and LCM of 6, 72 and 120, using the prime factorisation method.

Exercise 1.1

Practice · 15
  1. Express each number as a product of its prime factors:
    Ex 1.1 Q1 (i)
    140
  2. Ex 1.1 Q1 (ii)
    156
  3. Ex 1.1 Q1 (iii)
    3825
  4. Ex 1.1 Q1 (iv)
    5005
  5. Ex 1.1 Q1 (v)
    7429
  6. Find the LCM and HCF of the following pairs of integers and verify that LCM ×\times HCF = product of the two numbers.
    Ex 1.1 Q2 (i)
    26 and 91
  7. Ex 1.1 Q2 (ii)
    510 and 92
  8. Ex 1.1 Q2 (iii)
    336 and 54
  9. Find the LCM and HCF of the following integers by applying the prime factorisation method.
    Ex 1.1 Q3 (i)
    12, 15 and 21
  10. Ex 1.1 Q3 (ii)
    17, 23 and 29
  11. Ex 1.1 Q3 (iii)
    8, 9 and 25
  12. Ex 1.1 Q4
    Given that HCF(306,657)=9\text{HCF}(306, 657) = 9, find LCM(306,657)\text{LCM}(306, 657).
  13. Ex 1.1 Q5
    Check whether 6n6^n can end with the digit 0 for any natural number nn.
  14. Ex 1.1 Q6
    Explain why 7×11×13+137 \times 11 \times 13 + 13 and 7×6×5×4×3×2×1+57 \times 6 \times 5 \times 4 \times 3 \times 2 \times 1 + 5 are composite numbers.
  15. Ex 1.1 Q7
    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?

1.3 Revisiting Irrational Numbers

8 q

Solved Examples

Worked · 3
  1. 1.2 Eg.5
    Prove that 3\sqrt{3} is irrational.
  2. 1.2 Eg.6
    Show that 5−35 - \sqrt{3} is irrational.
  3. 1.2 Eg.7
    Show that 323\sqrt{2} is irrational.

Exercise 1.2

Practice · 5
  1. Ex 1.2 Q1
    Prove that 5\sqrt{5} is irrational.
  2. Ex 1.2 Q2
    Prove that 3+253 + 2\sqrt{5} is irrational.
  3. Prove that the following are irrationals :
    Ex 1.2 Q3 (i)
    12\frac{1}{\sqrt{2}}
  4. Ex 1.2 Q3 (ii)
    757\sqrt{5}
  5. Ex 1.2 Q3 (iii)
    6+26 + \sqrt{2}