Mathematics NCERT Grade 10, Chapter 1, Real numbers are values that are used to represent continuous quantity. The first section starts with the introduction of real numbers and the two important properties of real numbers namely:
1. Euclid's division algorithm, as the name suggests, it is related to the divisibility of integers.
It states that a positive integer 'a' can be divided by another positive integer 'b' in such a way that it leaves a remainder 'r' that is smaller than 'b'.
2. The fundamental theorem of arithmetic: It deals with the multiplication of positive integers.
• Every composite number can be expressed as a product of primes in a unique way.
This theorem has significant applications in mathematics. The two main applications of the fundamental theorem of arithmetic are cited in this section:
a. To prove the irrationality of numbers such as, $\sqrt{2}$$\sqrt{3}$ and $\sqrt{5}$ .
b. To explore when exactly the decimal expansion of a rational number is terminating and when it is not terminating.
Various solved examples are given to make the problems more understandable.  Section 1.4 explains the topic of revisiting irrational numbers. A number is irrational if it cannot be written in the form of p/q where q = 0. Two theorems are explained which will help in proving the irrationality of numbers. A few examples are shown that how a number is proven irrational.  A short exercise is given for the assessment of students. The last section comprises of the topic- Revisiting rational numbers and their decimal expansion. Two types of decimal expansion are discussed.
• Repeating decimal expansion
• Non-repeating decimal expansion
• In this section, three theorems are included and in the last theorem it is concluded that "the decimal expansion of every rational number is either terminating or non-terminating repeating".  In the last exercise,1.4 certain problems are given in which the nature of decimal expansion is to be determined without using the long division method. Finally, the chapter ends with a summary.

#### Question 1:

Use Euclid’s division algorithm to find the HCF of:

(i) 135 and 225

Since 225 > 135, we apply the division lemma to 225 and 135 to obtain

225 = 135 × 1 + 90

Since remainder 90 ≠ 0, we apply the division lemma to 135 and 90 to obtain

135 = 90 × 1 + 45

We consider the new divisor 90 and new remainder 45, and apply the division lemma to obtain

90 = 2 × 45 + 0

Since the remainder is zero, the process stops.

Since the divisor at this stage is 45,

Therefore, the HCF of 135 and 225 is 45.

(ii)196 and 38220

Since 38220 > 196, we apply the division lemma to 38220 and 196 to obtain

38220 = 196 × 195 + 0

Since the remainder is zero, the process stops.

Since the divisor at this stage is 196,

Therefore, HCF of 196 and 38220 is 196.

(iii)867 and 255

Since 867 > 255, we apply the division lemma to 867 and 255 to obtain

867 = 255 × 3 + 102

Since remainder 102 ≠ 0, we apply the division lemma to 255 and 102 to obtain

255 = 102 × 2 + 51

We consider the new divisor 102 and new remainder 51, and apply the division lemma to obtain

102 = 51 × 2 + 0

Since the remainder is zero, the process stops.

Since the divisor at this stage is 51,

Therefore, HCF of 867 and 255 is 51.

##### Video Solution for real numbers (Page: 7 , Q.No.: 1)

NCERT Solution for Class 10 math - real numbers 7 , Question 1

#### Question 2:

Show that any positive odd integer is of the form , or , or , where q is some integer.

Let a be any positive integer and b = 6. Then, by Euclid’s algorithm,

a = 6q + rfor some integer q0, and r = 0, 1, 2, 3, 4, 5 because 0 r < 6.

Therefore, a = 6q or 6q + 1 or 6q + 2 or 6q + 3 or 6q + 4 or 6q + 5

Also, 6q + 1 = 2 × 3q + 1 = 2k1 + 1, where k1 is a positive integer

6q + 3 = (6q + 2) + 1 = 2 (3q + 1) + 1 = 2k2 + 1, where k2 is an integer

6q + 5 = (6q + 4) + 1 = 2 (3q + 2) + 1 = 2k3 + 1, where k3 is an integer

Clearly, 6q + 1, 6q + 3, 6q + 5 are of the form 2k + 1, where k is an integer.

Therefore, 6q + 1, 6q + 3, 6q + 5 are not exactly divisible by 2. Hence, these expressions of numbers are odd numbers.

And therefore, any odd integer can be expressed in the form 6q + 1, or 6q + 3,

or 6q + 5

##### Video Solution for real numbers (Page: 7 , Q.No.: 2)

NCERT Solution for Class 10 math - real numbers 7 , Question 2

#### Question 3:

An army contingent of 616 members is to march behind an army band of 32 members in a parade. The two groups are to march in the same number of columns. What is the maximum number of columns in which they can march?

HCF (616, 32) will give the maximum number of columns in which they can march.

We can use Euclid’s algorithm to find the HCF.

616 = 32 × 19 + 8

32 = 8 × 4 + 0

The HCF (616, 32) is 8.

Therefore, they can march in 8 columns each.

##### Video Solution for real numbers (Page: 7 , Q.No.: 3)

NCERT Solution for Class 10 math - real numbers 7 , Question 3

#### Question 4:

Use Euclid’s division lemma to show that the square of any positive integer is either of form 3m or 3m + 1 for some integer m.

[Hint: Let x be any positive integer then it is of the form 3q, 3q + 1 or 3q + 2. Now square each of these and show that they can be rewritten in the form 3m or 3m + 1.]

Let a be any positive integer and b = 3.

Then a = 3q + r for some integer q ≥ 0

And r = 0, 1, 2 because 0 ≤ r < 3

Therefore, a = 3q or 3q + 1 or 3q + 2

Or,

Where k1, k2, and k3 are some positive integers

Hence, it can be said that the square of any positive integer is either of the form 3m or 3m + 1.

##### Video Solution for real numbers (Page: 7 , Q.No.: 4)

NCERT Solution for Class 10 math - real numbers 7 , Question 4

#### Question 5:

Use Euclid’s division lemma to show that the cube of any positive integer is of the form 9m, 9m + 1 or 9m + 8.

Let a be any positive integer and b = 3

a = 3q + r, where q ≥ 0 and 0 ≤ r < 3

Therefore, every number can be represented as these three forms. There are three cases.

Case 1: When a = 3q,

Where m is an integer such that m =

Case 2: When a = 3q + 1,

a3 = (3q +1)3

a3 = 27q3 + 27q2 + 9q + 1

a3 = 9(3q3 + 3q2 + q) + 1

a3 = 9m + 1

Where m is an integer such that m = (3q3 + 3q2 + q)

Case 3: When a = 3q + 2,

a3 = (3q +2)3

a3 = 27q3 + 54q2 + 36q + 8

a3 = 9(3q3 + 6q2 + 4q) + 8

a3 = 9m + 8

Where m is an integer such that m = (3q3 + 6q2 + 4q)

Therefore, the cube of any positive integer is of the form 9m, 9m + 1,
or 9m + 8.

##### Video Solution for real numbers (Page: 7 , Q.No.: 5)

NCERT Solution for Class 10 math - real numbers 7 , Question 5

#### Question 1:

Express each number as product of its prime factors:

##### Video Solution for real numbers (Page: 11 , Q.No.: 1)

NCERT Solution for Class 10 math - real numbers 11 , Question 1

#### Question 2:

Find the LCM and HCF of the following pairs of integers and verify that LCM × HCF = product of the two numbers.

Hence, product of two numbers = HCF × LCM

Hence, product of two numbers = HCF × LCM

Hence, product of two numbers = HCF × LCM
##### Video Solution for real numbers (Page: 11 , Q.No.: 2)

NCERT Solution for Class 10 math - real numbers 11 , Question 2

#### Question 3:

Find the LCM and HCF of the following integers by applying the prime factorisation method.

##### Video Solution for real numbers (Page: 11 , Q.No.: 3)

NCERT Solution for Class 10 math - real numbers 11 , Question 3

#### Question 4:

Given that HCF (306, 657) = 9, find LCM (306, 657).

##### Video Solution for real numbers (Page: 11 , Q.No.: 4)

NCERT Solution for Class 10 math - real numbers 11 , Question 4

#### Question 5:

Check whether 6n can end with the digit 0 for any natural number n.

If any number ends with the digit 0, it should be divisible by 10 or in other words, it will also be divisible by 2 and 5 as 10 = 2 × 5

Prime factorisation of 6n = (2 ×3)n

It can be observed that 5 is not in the prime factorisation of 6n.

Hence, for any value of n, 6n will not be divisible by 5.

Therefore, 6n cannot end with the digit 0 for any natural number n.

##### Video Solution for real numbers (Page: 11 , Q.No.: 5)

NCERT Solution for Class 10 math - real numbers 11 , Question 5

#### Question 6:

Explain why 7 × 11 × 13 + 13 and 7 × 6 × 5 × 4 × 3 × 2 × 1 + 5 are composite numbers.

Numbers are of two types - prime and composite. Prime numbers can be divided by 1 and only itself, whereas composite numbers have factors other than 1 and itself.

It can be observed that

7 × 11 × 13 + 13 = 13 × (7 × 11 + 1) = 13 × (77 + 1)

= 13 × 78

= 13 ×13 × 6

The given expression has 6 and 13 as its factors. Therefore, it is a composite number.

7 × 6 × 5 × 4 × 3 × 2 × 1 + 5 = 5 ×(7 × 6 × 4 × 3 × 2 × 1 + 1)

= 5 × (1008 + 1)

= 5 ×1009

1009 cannot be factorised further. Therefore, the given expression has 5 and 1009 as its factors. Hence, it is a composite number.

##### Video Solution for real numbers (Page: 11 , Q.No.: 6)

NCERT Solution for Class 10 math - real numbers 11 , Question 6

#### Question 7:

There is a circular path around a sports field. Sonia takes 18 minutes to drive one round of the field, while Ravi takes 12 minutes for the same. Suppose they both start at the same point and at the same time, and go in the same direction. After how many minutes will they meet again at the starting point?

It can be observed that Ravi takes lesser time than Sonia for completing 1 round of the circular path. As they are going in the same direction, they will meet again at the same time when Ravi will have completed 1 round of that circular path with respect to Sonia. And the total time taken for completing this 1 round of circular path will be the LCM of time taken by Sonia and Ravi for completing 1 round of circular path respectively i.e., LCM of 18 minutes and 12 minutes.

18 = 2 × 3 × 3

And, 12 = 2 × 2 × 3

LCM of 12 and 18 = 2 × 2 × 3 × 3 = 36

Therefore, Ravi and Sonia will meet together at the starting pointafter 36 minutes.

##### Video Solution for real numbers (Page: 11 , Q.No.: 7)

NCERT Solution for Class 10 math - real numbers 11 , Question 7

#### Question 1:

Prove that is irrational.

Let is a rational number.

Therefore, we can find two integers a, b (b ≠ 0) such that

Let a and b have a common factor other than 1. Then we can divide them by the common factor, and assume that a and b are co-prime.

Therefore, a2 is divisible by 5 and it can be said that a is divisible by 5.

Let a = 5k, where k is an integer

This means that b2 is divisible by 5 and hence, b is divisible by 5.

This implies that a and b have 5 as a common factor.

And this is a contradiction to the fact that a and b are co-prime.

Hence,cannot be expressed as or it can be said that is irrational.

##### Video Solution for real numbers (Page: 14 , Q.No.: 1)

NCERT Solution for Class 10 math - real numbers 14 , Question 1

#### Question 2:

Prove that is irrational.

Let is rational.

Therefore, we can find two integers a, b (b ≠ 0) such that

Since a and b are integers, will also be rational and therefore,is rational.

This contradicts the fact that is irrational. Hence, our assumption that is rational is false. Therefore, is irrational.

##### Video Solution for real numbers (Page: 14 , Q.No.: 2)

NCERT Solution for Class 10 math - real numbers 14 , Question 2

#### Question 3:

Prove that the following are irrationals:

Let is rational.

Therefore, we can find two integers a, b (b ≠ 0) such that

is rational as a and b are integers.

Therefore, is rational which contradicts to the fact that is irrational.

Hence, our assumption is false and is irrational.

Let is rational.

Therefore, we can find two integers a, b (b ≠ 0) such that

for some integers a and b

is rational as a and b are integers.

Therefore, should be rational.

This contradicts the fact thatis irrational. Therefore, our assumption that is rational is false. Hence, is irrational.

Let be rational.

Therefore, we can find two integers a, b (b ≠ 0) such that

Since a and b are integers, is also rational and hence, should be rational. This contradicts the fact that is irrational. Therefore, our assumption is false and hence, is irrational.

##### Video Solution for real numbers (Page: 14 , Q.No.: 3)

NCERT Solution for Class 10 math - real numbers 14 , Question 3

#### Question 1:

Without actually performing the long division, state whether the following rational numbers will have a terminating decimal expansion or a non-terminating repeating decimal expansion:

(i)

The denominator is of the form 5m.

Hence, the decimal expansion ofis terminating.

(ii)

The denominator is of the form 2m.

Hence, the decimal expansion of is terminating.

(iii)

455 = 5 × 7 × 13

Since the denominator is not in the form 2m × 5n, and it also contains 7 and 13 as its factors, its decimal expansion will be non-terminating repeating.

(iv)

1600 = 26 × 52

The denominator is of the form 2m × 5n.

Hence, the decimal expansion of is terminating.

(v)

Since the denominator is not in the form 2m × 5n, and it has 7 as its factor, the decimal expansion of is non-terminating repeating.

(vi)

The denominator is of the form 2m × 5n.

Hence, the decimal expansion of is terminating.

(vii)

Since the denominator is not of the form 2m × 5n, and it also has 7 as its factor, the decimal expansion of is non-terminating repeating.

(viii)

The denominator is of the form 5n.

Hence, the decimal expansion of is terminating.

(ix)

The denominator is of the form 2m × 5n.

Hence, the decimal expansion of is terminating.

(x)

Since the denominator is not of the form 2m × 5n, and it also has 3 as its factors, the decimal expansion of is non-terminating repeating.

##### Video Solution for real numbers (Page: 17 , Q.No.: 1)

NCERT Solution for Class 10 math - real numbers 17 , Question 1

#### Question 2:

Write down the decimal expansions of those rational numbers in Question 1 above which have terminating decimal expansions.

(viii)

##### Video Solution for real numbers (Page: 18 , Q.No.: 2)

NCERT Solution for Class 10 math - real numbers 18 , Question 2

#### Question 3:

The following real numbers have decimal expansions as given below. In each case, decide whether they are rational or not. If they are rational, and of the form , what can you say about the prime factor of q?

(i) 43.123456789 (ii) 0.120120012000120000… (iii)

(i) 43.123456789

Since this number has a terminating decimal expansion, it is a rational number of the form and q is of the form

i.e., the prime factors of q will be either 2 or 5 or both.

(ii) 0.120120012000120000 …

The decimal expansion is neither terminating nor recurring. Therefore, the given number is an irrational number.

(iii)

Since the decimal expansion is non-terminating recurring, the given number is a rational number of the form and q is not of the form i.e., the prime factors of q will also have a factor other than 2 or 5.

##### Video Solution for real numbers (Page: 18 , Q.No.: 3)

NCERT Solution for Class 10 math - real numbers 18 , Question 3

#### Question 17:

What is the least number that divisible by all the natural numbers from 1 to 10 (both inclusive)?

(a) 100
(b) 1260
(c) 2520
(d) 5040

(c) 2520

We have to find the least number that is divisible by all numbers from 1 to 10.
∴ LCM (1 to 10) =
Thus, 2520 is the least number that is divisible by every element and is equal to the least common multiple.

#### Question 21:

a and b are two positive integers such that the least prime factor of a is 3 and the least prime factor of b is 5. Then, the least prime factor of (a + b) is

(a) 2
(b) 3
(c) 5
(d) 8

(a) 2

Since 5 + 3 = 8, the least prime factor of a + b has to be 2, unless a + b is a prime number greater than 2.
If a + b is a prime number greater than 2, then a + must be an odd number. So, either a or b must be an even number. If a is even, then the least prime factor of is 2, which is not 3 or 5. So, neither a nor b can be an even number. Hence, a + b cannot be a prime number greater than 2 if the least prime factor of a is 3 or 5.

#### Question 24:

Which of the following rational numbers is expressible as a terminating decimal?

(a) $\frac{124}{165}$
(b) $\frac{131}{30}$
(c) $\frac{2027}{625}$
(d) $\frac{1625}{462}$

(c) $\frac{2027}{625}$

; we know 5 and 33 are not the factors of 124. It is in its simplest form and it cannot be expressed as the product of  for some non-negative integers .

So, it cannot be expressed as a terminating decimal.

$\frac{131}{30}$ = ; we know 5 and 6 are not the factors of 131. Its is in its simplest form and it cannot be expressed as the product of ( ) for some non-negative integers .

So, it cannot be expressed as a terminating decimal.

; as it is of the form , where  are non-negative integers.
So, it is a terminating decimal.

; we know 2, 7 and 33 are not the factors of 1625. It is in its simplest form and cannot be expressed as the product of  for some non-negative integers $m,n$.
So, it cannot be expressed as a terminating decimal.

#### Question 35:

$\left(2+\sqrt{2}\right)$ is
(a) an integer
(b) a rational number
(c) an irrational number
(d) none of these

∴​  = $\sqrt{2}$ = irrational