Class 10 Math CBSE Chapter 1 Real Numbers – Complete Guide with NCERT Solutions

Introduction

Real Numbers form the foundation of higher mathematics. In Class 9, you learned about rational and irrational numbers. In Class 10, this chapter revisits Euclid’s Division Algorithm, the Fundamental Theorem of Arithmetic, and proofs of irrationality. You’ll also explore HCF, LCM, and decimal expansions using prime factorisation.

Key Formulas and Theorems

• Euclid’s Division Lemma: For any integers $a,b$ (with $a>b>0$), there exist unique integers $q,r$ such that

$$\mathrm{<b><span\; style="font-family:\; verdana;">a</span></b>}<b><span\; style="font-family:\; verdana;">=</span></b>\mathrm{<b><span\; style="font-family:\; verdana;">b</span></b>}\mathrm{<b><span\; style="font-family:\; verdana;">q</span></b>}<b><span\; style="font-family:\; verdana;">+</span></b>\mathrm{<b><span\; style="font-family:\; verdana;">r</span></b>}<b><span\; style="font-family:\; verdana;">,</span></b>\mathrm{<b><span\; style="font-family:\; verdana;">0</span></b>}<b><span\; style="font-family:\; verdana;">\le </span></b>\mathrm{<b><span\; style="font-family:\; verdana;">r</span></b>}<b><span\; style="font-family:\; verdana;"><</span></b>\mathrm{<b><span\; style="font-family:\; verdana;">b</span></b>}$$

• Fundamental Theorem of Arithmetic: Every composite number can be expressed uniquely as a product of primes (order of factors doesn’t matter).

• Relation between HCF and LCM:

$$\mathrm{<b><span\; style="font-family:\; verdana;">H</span></b>}\mathrm{<b><span\; style="font-family:\; verdana;">C</span></b>}\mathrm{<b><span\; style="font-family:\; verdana;">F</span></b>}<b><span\; style="font-family:\; verdana;">(</span></b>\mathrm{<b><span\; style="font-family:\; verdana;">a</span></b>}<b><span\; style="font-family:\; verdana;">,</span></b>\mathrm{<b><span\; style="font-family:\; verdana;">b</span></b>}<b><span\; style="font-family:\; verdana;">)</span></b><b><span\; style="font-family:\; verdana;">\times </span></b>\mathrm{<b><span\; style="font-family:\; verdana;">L</span></b>}\mathrm{<b><span\; style="font-family:\; verdana;">C</span></b>}\mathrm{<b><span\; style="font-family:\; verdana;">M</span></b>}<b><span\; style="font-family:\; verdana;">(</span></b>\mathrm{<b><span\; style="font-family:\; verdana;">a</span></b>}<b><span\; style="font-family:\; verdana;">,</span></b>\mathrm{<b><span\; style="font-family:\; verdana;">b</span></b>}<b><span\; style="font-family:\; verdana;">)</span></b><b><span\; style="font-family:\; verdana;">=</span></b>\mathrm{<b><span\; style="font-family:\; verdana;">a</span></b>}<b><span\; style="font-family:\; verdana;">\times </span></b>\mathrm{<b><span\; style="font-family:\; verdana;">b</span></b>}$$

Solved NCERT Examples (Step‑by‑Step)

Example 1:

Question: Check whether $4^n$ ends with digit 0 for any natural number $n$. Solution:

• $4^n=(2^2{)}^n=2^{2n}$.

• Prime factorisation contains only 2.

• For a number to end with 0, it must be divisible by 5.

• Since 5 is not a factor, no value of $n$ makes $4^n$ end with 0.

Example 2:

Question: Find HCF and LCM of 6 and 20. Solution:

• $6=2\times 3$, $20=2^2\times 5$.

• HCF = product of smallest powers of common primes = $2$.

• LCM = product of greatest powers = $2^2\times 3\times 5=60$.

• Verification: $HCF\times LCM=2\times 60=120=6\times 20$.

Example 3:

Question: Find HCF and LCM of 96 and 404. Solution:

• $96=2^5\times 3$, $404=2^2\times 101$.

• HCF = $2^2=4$.

• LCM = $\frac{96\times 404}{4}=9696$.

Example 4:

Question: Find HCF and LCM of 6, 72, and 120. Solution:

• $6=2\times 3$, $72=2^3\times 3^2$, $120=2^3\times 3\times 5$.

• HCF = $2\times 3=6$.

• LCM = $2^3\times 3^2\times 5=360$.

Example 5:

Question: Prove $\sqrt{2}$ is irrational. Solution:

• Assume $\sqrt{2}=\frac{a}{b}$, where $a,b$ are coprime.

• Then $2b^2=a^2$.

• So $a$ divisible by 2 → let $a=2c$.

• Substituting: $2b^2=4c^2\Rightarrow b^2=2c^2$.

• So $b$ divisible by 2.

• Contradiction: $a,b$ not coprime.

• Hence, $\sqrt{2}$ is irrational.

FAQs with Solutions

Here are 15 FAQs based on NCERT exercises:

1. Express 3825 as product of primes. $3825=3\times 5^2\times 17\times 15$.

2. Find HCF and LCM of 26 and 91. $26=2\times 13$, $91=7\times 13$. HCF = 13, LCM = $2\times 7\times 13=182$.

3. Find HCF and LCM of 510 and 92. $510=2\times 3\times 5\times 17$, $92=2^2\times 23$. HCF = 2, LCM = 11730.

4. Find HCF and LCM of 336 and 54. $336=2^4\times 3\times 7$, $54=2\times 3^3$. HCF = 6, LCM = 3024.

5. Find HCF and LCM of 12, 15, 21. HCF = 3, LCM = 420.

6. Find HCF and LCM of 17, 23, 29. All primes → HCF = 1, LCM = 11339.

7. Find HCF and LCM of 8, 9, 25. HCF = 1, LCM = 1800.

8. Given HCF(306, 657) = 9, find LCM. LCM = $\frac{306\times 657}{9}=22338$.

9. Check whether $6^n$ ends with 0. Prime factors are only 2 and 3 → not divisible by 5 → cannot end with 0.

10. Explain why $7\times 11\times 13+13$ is composite. Expression = $13(7\times 11+1)$ → divisible by 13 → composite.

11. Explain why $7\times 6\times 5\times 4\times 3\times 2\times 1+5$ is composite. Expression = divisible by 5 → composite.

12. Sonia takes 18 min, Ravi 12 min around a track. When will they meet? LCM(18, 12) = 36 minutes.

13. Prove $\sqrt{3}$ is irrational. Similar contradiction method as $\sqrt{2}$.

14. Show $5-\sqrt{3}$ is irrational. Assume rational → leads to contradiction → irrational.

15. Show $3\sqrt{2}$ is irrational. Assume rational → contradiction → irrational.

Conclusion

This chapter builds the foundation for number theory and proofs in mathematics. Understanding prime factorisation, HCF, LCM, and irrationality proofs will help you in advanced topics.

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