Class 10 Maths NCERT Chapter 1 (Real Numbers) Solutions

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August 5, 2025
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Summary (What we have discussed)

Hello! Are you using the NCERT textbook to study Class 10 Maths Chapter 1: “Real Numbers“? The Fundamental Theorem of Arithmetic, Euclid’s Division Lemma, and the distinction between rational and irrational numbers are just a few of the fascinating questions that are covered in this page.

During your studies, you will discover things such as:

How to determine a number’s LCM (Least Common Multiple) and HCF (Highest Common Factor).
How to demonstrate why certain numbers, such as √2 and √3, are irrational.
How to determine if a decimal is non-terminating or terminating.

Class 10	MATHS
Chapter	REAL NUMBERS
NCERT MATHS (ENGLISH)	LONG ANSWER TYPE QUESTIONS
Medium	English
Academic Year	2023-2024

Question 1

For some integer m, every even integer is of the form

Question 2

For some integer q, every odd integer is of the form

Question 3

n²−1 is divisible by 8, if n is

Question 4

If HCF of 65 and 117 is expressible in the form 65m−117, then the value of m is

Question 5

The largest number which divides 70 and 125, leaving remainder 5 and 8 respectively, is

Question 6

If two positive integers m and n are expressible in the form m=pq³ and n=p³q² , where p, q are prime numbers, then HCF (m, n)=

Question 7

If two positive integers a and b are expressible in the form a=pq² and b=p³q ; p, q being prime numbers, then LCM (a, b) is

Question 8

The product of a non-zero rational number with an irrational number is always a/an

Question 9

What is the least number that is divisible by all the numbers 1 to 10

Question 10

The decimal expansion of the rational number 14587/1250 will terminate after

Question 11

Write whether every positive integer can be of the form 4q+2 where q is an integer, Justify your answer

Question 12

The product of two consecutive integers is divisible by 2. Is this statement true or false. Give Reason?

Question 13

The product of any three consecutive natural numbers is divisible by 6 (True/false).

Question 14

Write whether the square of any positive integer can be of the form 3m+2, where m is a natural number. Justify answer.

Question 15

A positive integer is the form of 3q+1 q, being a natural number. Can you write its square in any form other than 3m+1 i.e. 3m or 3m+2 for some integer? Justify your answer.

Question 16

The number 525 and 3000 are both divisible only by 3,5,15,25,75. What is HCF (525, 3000)? Justify your answer.

Question 17

Explain why 3×5×7+7 is a composite number.

Question 18

Can two number have 18 as their HCF and 380 as their LCM? Give reason

Question 19

Without actually performing the long divison, find if 987/10500 will have terminating or non-terminating (repeating) decimal expansion. Give reasons for your answer

Question 20

A rational number in its decimal expansion is 327.7081. What can you say about the prime factors of q, when this number is expressed in the from p/q ? Give reason

Question 21

Prove that the square of any positive integer is of the form 4q or 4q+1 for some integer q .

Question 22

Show that cube of any positive integer is of the form 4m, 4m+1 or 4m+3, for some integer m.

Question 23

Show that the square of any positive integer cannot be of the form 5q+2 or 5q+3 for some integer q.

Question 24

Show that the square of any positive integer cannot be of the form 6m+2 or 6m+5 for some integer q.

Question 25

Show that the square of any odd integer is of the form 4m+1, for some integer m.

Question 26

If n is an odd positive integer, show that (n²−1) is divisible by 8.

Question 27

Prove that if x and y are odd positive integers, then x²+y² is even but not divisible by 4.

Question 28

Use Euclid division algorithm to find the HCF of 441, 567 and 693.

Question 29

Using Euclid’s division algorithm, find the largest number that divides 1251, 9377 and 15628 leaving remainders 1, 2 and 3, respectively.

Question 30

Prove that √3+√5 is irrational

Question 31

Show that 12ⁿ cannot end with the digits 0 or 5 for any natural number n

Question 32

In a morning walk, three persons step off together and their steps measure 40cm,42cm and 45cm, respectively. What is the minimum distance each should walk so that each can cover the same distance in complete steps?

Question 33

Write the denominator of the rational number 257/5000 in the form 2^m×5ⁿ, where m, n and non-negative integers. Hence, write its decimal expansion without actual division.

Question 34

Prove that √p+√q is an irrational, where pandq are primes.

Question 35

Show that the cube of a positive integer of the form 6q+r,q is an integer and r=0,1,2,3,4,5 is also of the form 6m+r

Question 36

Show that one and only one out of n,n+2or,n+4 is divisible by 3, where n is any positive integer.

Question 37

Prove that one of every three consecutive positive integers is divisible by 3.

Question 38

For any positive integer n , prove that n³−n is divisible by 6.

Question 39

Show that one and only one out of n,n+4,n+8,n+12andn+16 is divisible by 5, where n is any positive integer.

Class 10	MATHS
Chapter	REAL NUMBERS
NCERT MATHS (ENGLISH)	LONG ANSWER TYPE QUESTIONS
Medium	English
Academic Year	2023-2024

Question 1

For some integer m, every even integer is of the form

Question 2

For some integer q, every odd integer is of the form

Question 3

n²−1 is divisible by 8, if n is

Question 4

If HCF of 65 and 117 is expressible in the form 65m−117, then the value of m is

Question 5

The largest number which divides 70 and 125, leaving remainder 5 and 8 respectively, is

Question 6

If two positive integers m and n are expressible in the form m=pq³ and n=p³q² , where p, q are prime numbers, then HCF (m, n)=

Question 7

If two positive integers a and b are expressible in the form a=pq² and b=p³q ; p, q being prime numbers, then LCM (a, b) is

Question 8

The product of a non-zero rational number with an irrational number is always a/an

Question 9

What is the least number that is divisible by all the numbers 1 to 10

Question 10

The decimal expansion of the rational number 14587/1250 will terminate after

Question 11

Write whether every positive integer can be of the form 4q+2 where q is an integer, Justify your answer

Question 12

The product of two consecutive integers is divisible by 2. Is this statement true or false. Give Reason?

Question 13

The product of any three consecutive natural numbers is divisible by 6 (True/false).

Question 14

Write whether the square of any positive integer can be of the form 3m+2, where m is a natural number. Justify answer.

Question 15

A positive integer is the form of 3q+1 q, being a natural number. Can you write its square in any form other than 3m+1 i.e. 3m or 3m+2 for some integer? Justify your answer.

Question 16

The number 525 and 3000 are both divisible only by 3,5,15,25,75. What is HCF (525, 3000)? Justify your answer.

Question 17

Explain why 3×5×7+7 is a composite number.

Question 18

Can two number have 18 as their HCF and 380 as their LCM? Give reason

Question 19

Without actually performing the long divison, find if 987/10500 will have terminating or non-terminating (repeating) decimal expansion. Give reasons for your answer

Question 20

A rational number in its decimal expansion is 327.7081. What can you say about the prime factors of q, when this number is expressed in the from p/q ? Give reason

Question 21

Prove that the square of any positive integer is of the form 4q or 4q+1 for some integer q .

Question 22

Show that cube of any positive integer is of the form 4m, 4m+1 or 4m+3, for some integer m.

Question 23

Show that the square of any positive integer cannot be of the form 5q+2 or 5q+3 for some integer q.

Question 24

Show that the square of any positive integer cannot be of the form 6m+2 or 6m+5 for some integer q.

Question 25

Show that the square of any odd integer is of the form 4m+1, for some integer m.

Question 26

If n is an odd positive integer, show that (n²−1) is divisible by 8.

Question 27

Prove that if x and y are odd positive integers, then x²+y² is even but not divisible by 4.

Question 28

Use Euclid division algorithm to find the HCF of 441, 567 and 693.

Question 29

Using Euclid’s division algorithm, find the largest number that divides 1251, 9377 and 15628 leaving remainders 1, 2 and 3, respectively.

Question 30

Prove that √3+√5 is irrational

Question 31

Show that 12ⁿ cannot end with the digits 0 or 5 for any natural number n

Question 32

In a morning walk, three persons step off together and their steps measure 40cm,42cm and 45cm, respectively. What is the minimum distance each should walk so that each can cover the same distance in complete steps?

Question 33

Write the denominator of the rational number 257/5000 in the form 2^m×5ⁿ, where m, n and non-negative integers. Hence, write its decimal expansion without actual division.

Question 34

Prove that √p+√q is an irrational, where pandq are primes.

Question 35

Show that the cube of a positive integer of the form 6q+r,q is an integer and r=0,1,2,3,4,5 is also of the form 6m+r

Question 36

Show that one and only one out of n,n+2or,n+4 is divisible by 3, where n is any positive integer.

Question 37

Prove that one of every three consecutive positive integers is divisible by 3.

Question 38

For any positive integer n , prove that n³−n is divisible by 6.

Question 39

Show that one and only one out of n,n+4,n+8,n+12andn+16 is divisible by 5, where n is any positive integer.

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