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Mathematics · Real Numbers

NCERT Examples

All 7 worked examples from Real Numbers, with independently verified solutions and direct navigation.

Mathematics · Chapter 1 · NCERT Examples

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NCERT Example

Example 1

prime factorisation · terminal digit

Consider the numbers 4ⁿ, where n is a natural number. Check whether there is any value of n for which 4ⁿ ends with the digit zero.

Solution

4n=(22)n=22n4^n=(2^2)^n=2^{2n}

A number ending in zero is divisible by 10 = 2 × 5, so its prime factorisation contains both 2 and 5. The factorisation of 4ⁿ contains only 2 and has no factor 5.

Final answer

There is no natural number n for which 4ⁿ ends with zero.

NCERT Example

Example 2

HCF · LCM · prime factorisation

Find the LCM and HCF of 6 and 20 by the prime factorisation method.

Solution

6=2×3,20=22×56=2\times3,\qquad20=2^2\times5
HCF⁡(6,20)=2\operatorname{HCF}(6,20)=2
LCM⁡(6,20)=22×3×5=60\operatorname{LCM}(6,20)=2^2\times3\times5=60

Final answer

HCF = 2 and LCM = 60.

NCERT Example

Example 3

HCF · LCM · prime factorisation

Find the HCF of 96 and 404 by the prime factorisation method. Hence, find their LCM.

Solution

96=25×3,404=22×10196=2^5\times3,\qquad404=2^2\times101
HCF⁡(96,404)=22=4\operatorname{HCF}(96,404)=2^2=4
LCM⁡(96,404)=96×4044=9696\operatorname{LCM}(96,404)=\frac{96\times404}{4}=9696

Final answer

HCF = 4 and LCM = 9696.

NCERT Example

Example 4

HCF · LCM · prime factorisation

Find the HCF and LCM of 6, 72 and 120, using the prime factorisation method.

Solution

6=2×3,72=23×32,120=23×3×56=2\times3,\quad72=2^3\times3^2,\quad120=2^3\times3\times5
HCF⁡=2×3=6\operatorname{HCF}=2\times3=6
LCM⁡=23×32×5=360\operatorname{LCM}=2^3\times3^2\times5=360

Final answer

HCF = 6 and LCM = 360.

NCERT Example

Example 5

irrational numbers · proof by contradiction

Prove that √3 is irrational.

Solution

Suppose √3 is rational and write it in lowest terms.

3=pq,gcd⁡(p,q)=1,q≠0\sqrt3=\frac pq,\qquad \gcd(p,q)=1,\quad q\ne0
p2=3q2p^2=3q^2

Thus 3 divides p², so 3 divides p. Put p = 3k. Substitution gives q² = 3k², so 3 also divides q. This contradicts p and q being coprime.

Final answer

Therefore, √3 is irrational.

NCERT Example

Example 6

irrational numbers · proof by contradiction

Show that 5 − √3 is irrational.

Solution

Suppose 5 − √3 = r for a rational number r.

3=5−r\sqrt3=5-r

The right side would be rational, contradicting the irrationality of √3.

Final answer

Therefore, 5 − √3 is irrational.

NCERT Example

Example 7

irrational numbers · proof by contradiction

Show that 3√2 is irrational.

Solution

Suppose 3√2 = r for a rational number r.

2=r3\sqrt2=\frac r3

The right side would be rational, contradicting the irrationality of √2.

Final answer

Therefore, 3√2 is irrational.