NCERT Solutions for Class 10 Mathematics Chapter 1 Real Numbers

Q1

State Euclid's division lemma and explain how it is used to find the HCF of two positive integers.

Euclid's division lemma says that for positive integers

a

and

b

, there exist unique integers

q

and

r

such that

a=bq+r

, where

0\le r<b

. Repeated use of this relation gives Euclid's division algorithm for finding HCF.

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Q2

Find the HCF of $135$ and $225$ using Euclid's division algorithm.

Using Euclid's algorithm:

225=135\times1+90

,

135=90\times1+45

,

90=45\times2+0

. Therefore, HCF

=45

.

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Q3

State the Fundamental Theorem of Arithmetic. How does it help in finding HCF and LCM?

The Fundamental Theorem of Arithmetic states that every composite number can be expressed as a product of primes, and this factorisation is unique apart from the order of the prime factors. HCF uses the lowest powers of common primes; LCM uses the highest powers of all primes.

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Q4

Use the Fundamental Theorem of Arithmetic to prove that $\sqrt{2}$ is irrational.

Assume

\sqrt{2}=\frac{p}{q}

in lowest terms. Then

p^2=2q^2

, so

p^2

is even and hence

p

is even. Let

p=2k

. Then

4k^2=2q^2

, so

q^2=2k^2

, making

q

even. This contradicts that

p

and

q

are coprime. Hence

\sqrt{2}

is irrational.

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Q5

When does the rational number $\frac{p}{q}$ have a terminating decimal expansion?

If

p

and

q

are coprime and the prime factorisation of

q

is of the form

2^m5^n

, then

\frac{p}{q}

has a terminating decimal expansion. Otherwise it has a non-terminating recurring decimal expansion.

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Q6

Decide whether $\frac{13}{3125}$ and $\frac{17}{6}$ have terminating decimal expansions.

For

\frac{13}{3125}

,

3125=5^5

, so it terminates. For

\frac{17}{6}

,

6=2\times3

contains prime factor

3

, so it is non-terminating recurring.

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Real Numbers

All questions

State Euclid's division lemma and explain how it is used to find the HCF of two positive integers.

Find the HCF of $135$ and $225$ using Euclid's division algorithm.

State the Fundamental Theorem of Arithmetic. How does it help in finding HCF and LCM?

Use the Fundamental Theorem of Arithmetic to prove that $\sqrt{2}$ is irrational.

When does the rational number $\frac{p}{q}$ have a terminating decimal expansion?

Decide whether $\frac{13}{3125}$ and $\frac{17}{6}$ have terminating decimal expansions.

Other Chapters

Real Numbers

All questions

State Euclid's division lemma and explain how it is used to find the HCF of two positive integers.

Find the HCF of 135135135 and 225225225 using Euclid's division algorithm.

State the Fundamental Theorem of Arithmetic. How does it help in finding HCF and LCM?

Use the Fundamental Theorem of Arithmetic to prove that 2\sqrt{2}2​ is irrational.

When does the rational number pq\frac{p}{q}qp​ have a terminating decimal expansion?

Decide whether 133125\frac{13}{3125}312513​ and 176\frac{17}{6}617​ have terminating decimal expansions.

Other Chapters

Find the HCF of $135$ and $225$ using Euclid's division algorithm.

Use the Fundamental Theorem of Arithmetic to prove that $\sqrt{2}$ is irrational.

When does the rational number $\frac{p}{q}$ have a terminating decimal expansion?

Decide whether $\frac{13}{3125}$ and $\frac{17}{6}$ have terminating decimal expansions.