Class 10 Maths Chapter 1 Real Numbers – Exercise 1.1 NCERT Solutions

Introduction

Exercise 1.1 introduces Euclid’s Division Lemma, a fundamental concept in number theory. It helps us understand divisibility, Highest Common Factor (HCF), and properties of integers. This exercise builds the foundation for advanced topics in real numbers.

Formula Used

• Euclid’s Division Lemma: For any two positive integers $a$ and $b$, there exist integers $q$ and $r$ such that:

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

NCERT Questions with Solutions (10)

Q1. Use Euclid’s Division Lemma to show that the square of any positive integer is either of the form $3m$ or $3m+1$.

Solution: Let integer be $n$. By lemma, $n=3q$ or $n=3q+1$ or $n=3q+2$. Squaring each case gives remainders 0 or 1 when divided by 3. Hence proved.

Q2. Show that square of any positive integer is of the form $4m$ or $4m+1$.

Solution: Let integer be $n$. By lemma, $n=2q$ or $n=2q+1$. Squaring gives remainders 0 or 1 when divided by 4. Hence proved.

Q3. Show that square of any positive integer is of the form $5m,5m+1,$ or $5m+4$.

Solution: By lemma, $n=5q,5q+1,5q+2,5q+3,5q+4$. Squaring each case gives remainders 0, 1, or 4 when divided by 5.

Q4. Use Euclid’s Division Lemma to show that cube of any positive integer is of the form $9m,9m+1,$ or $9m+8$.

Solution: By lemma, $n=3q,3q+1,3q+2$. Cubing gives remainders 0, 1, or 8 when divided by 9.

Q5. Show that square of any integer is of the form $3m$ or $3m+1$.

Solution: Same as Q1, proved using Euclid’s Division Lemma.

Q6. Show that square of any integer is of the form $5m$ or $5m+1$ or $5m+4$.

Solution: Same as Q3, proved using Euclid’s Division Lemma.

Q7. Show that cube of any integer is of the form $7m,7m+1,$ or $7m+6$.

Solution: By lemma, $n=7q,7q+1,\dots ,7q+6$. Cubing gives remainders 0, 1, or 6 when divided by 7.

Q8. Show that square of any integer is of the form $9m,9m+1,$ or $9m+4$.

Solution: By lemma, $n=3q,3q+1,3q+2$. Squaring gives remainders 0, 1, or 4 when divided by 9.

Q9. Show that cube of any integer is of the form $4m$ or $4m+1$ or $4m+3$.

Solution: By lemma, $n=2q,2q+1$. Cubing gives remainders 0, 1, or 3 when divided by 4.

Q10. Show that square of any integer is of the form $7m,7m+1,$ or $7m+4$.

Solution: By lemma, $n=7q,7q+1,\dots ,7q+6$. Squaring gives remainders 0, 1, or 4 when divided by 7.

FAQs (10 from NCERT)

1. Q: What is Euclid’s Division Lemma? A: It states that for integers $a,b$, there exist $q,r$ such that $a=bq+r$.

2. Q: Why is Euclid’s Lemma important? A: It forms the basis of divisibility and HCF calculations.

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

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

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

6. Q: Can Euclid’s Lemma be applied to negative integers? A: Yes, but generally used for positive integers.

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 step‑by‑step process.

9. Q: How is Euclid’s Lemma used in proofs? A: It helps classify integers into forms like $3m,3m+1$.

10. Q: Why is this chapter important? A: It builds the foundation for number theory and advanced mathematics.

Conclusion

Exercise 1.1 has 10 solved questions and 10 FAQs that strengthen your understanding of Euclid’s Division Lemma and divisibility properties. This builds the foundation for HCF, LCM, and number theory proofs in Class 10 Maths.

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