Class 10 Maths Chapter 1 Real Numbers – Exercise 1.2 NCERT Solutions

Introduction

Exercise 1.2 focuses on finding the HCF (Highest Common Factor) and LCM (Lowest Common Multiple) of numbers using Euclid’s Division Algorithm. This method is efficient and widely used in number theory and divisibility problems.

Formula Used

• Euclid’s Division Lemma:

$$a=bq+r,0\le r<b$$

• Relation between HCF and LCM:

$$HCF(a,b)\times LCM(a,b)=a\times b$$

NCERT Questions with Solutions (10)

Q1. 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. So, HCF = 45.

Q2. Find the HCF of 196 and 38220 using Euclid’s Algorithm.

Solution: 38220 ÷ 196 = 195 remainder 0. So, HCF = 196.

Q3. Find the HCF of 867 and 255 using Euclid’s Algorithm.

Solution: 867 ÷ 255 = 3 remainder 102. 255 ÷ 102 = 2 remainder 51. 102 ÷ 51 = 2 remainder 0. So, HCF = 51.

Q4. Find the HCF of 963 and 657 using Euclid’s Algorithm.

Solution: 963 ÷ 657 = 1 remainder 306. 657 ÷ 306 = 2 remainder 45. 306 ÷ 45 = 6 remainder 36. 45 ÷ 36 = 1 remainder 9. 36 ÷ 9 = 4 remainder 0. So, HCF = 9.

Q5. Find the HCF of 144 and 160 using Euclid’s Algorithm.

Solution: 160 ÷ 144 = 1 remainder 16. 144 ÷ 16 = 9 remainder 0. So, HCF = 16.

Q6. Find the HCF of 270 and 192 using Euclid’s Algorithm.

Solution: 270 ÷ 192 = 1 remainder 78. 192 ÷ 78 = 2 remainder 36. 78 ÷ 36 = 2 remainder 6. 36 ÷ 6 = 6 remainder 0. So, HCF = 6.

Q7. Find the HCF of 340 and 408 using Euclid’s Algorithm.

Solution: 408 ÷ 340 = 1 remainder 68. 340 ÷ 68 = 5 remainder 0. So, HCF = 68.

Q8. Find the HCF of 960 and 432 using Euclid’s Algorithm.

Solution: 960 ÷ 432 = 2 remainder 96. 432 ÷ 96 = 4 remainder 48. 96 ÷ 48 = 2 remainder 0. So, HCF = 48.

Q9. Find the HCF of 105 and 350 using Euclid’s Algorithm.

Solution: 350 ÷ 105 = 3 remainder 35. 105 ÷ 35 = 3 remainder 0. So, HCF = 35.

Q10. Find the HCF of 867 and 255, then use it to find LCM.

Solution: From Q3, HCF = 51.

$$LCM=\frac{867\times 255}{51}=4335$$

FAQs (10 from NCERT)

1. Q: What is Euclid’s Division Algorithm? A: A method to find HCF by repeated division.

2. Q: What is HCF? A: Highest Common Factor, the largest number dividing two integers.

3. Q: What is LCM? A: Lowest Common Multiple, the smallest number divisible by two integers.

4. Q: How is HCF related to LCM? A: $HCF\times LCM=Productofnumbers$.

5. Q: Why is Euclid’s Algorithm efficient? A: It reduces large numbers step by step until remainder is 0.

6. Q: Can Euclid’s Algorithm be applied to more than two numbers? A: Yes, by applying it pairwise.

7. Q: What is the remainder condition in Euclid’s Lemma? A: $0\le r<b$.

8. Q: What is the difference between Lemma and Algorithm? A: Lemma is a statement; algorithm is a process.

9. Q: Why is HCF important? A: It helps in simplifying fractions and divisibility problems.

10. Q: Why is LCM important? A: It helps in solving problems involving common multiples.

Conclusion

Exercise 1.2 has 10 solved questions and 10 FAQs that strengthen your understanding of HCF and LCM using Euclid’s Division Algorithm. This builds the foundation for divisibility and number theory in Class 10 Maths.

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