Class 10 Maths · Chapter 1

Real Numbers

82 Class 10 Maths questions on Real Numbers from the NCERT textbook, NCERT Exemplar and CBSE board papers 2026, each with its answer and a step-by-step solution.

CBSE Class 10 Maths previous year papers, with answers →

Questions · page 3 of 5

1. Q41 Exemplar · Exercise 1.3
   Show that $12^n$ cannot ​end ​with ​the ​digit 0 or 5 ​for ​any natural ​number $n$.
   Show answer

   Since $12^n=2^{2n}\times 3^n$ has no factor of 5, it is never divisible by 5, hence cannot end in 0 or 5.

   Step-by-step solution →
2. Q42 Exemplar · Exercise 1.3
   On ​a ​morning ​walk, three persons step off together and their steps ​measure ​40 cm, 42 cm and 45 cm, ​respectively. What is the minimum distance each should walk ​so that each can cover the same distance ​in complete steps?
   Show answer

   The minimum distance each should walk is $2520$ cm (i.e. 25.2 m).

   Step-by-step solution →
3. Q43 Exemplar · Exercise 1.3
   Write the denominator of ​the rational ​number $\frac{257}{5000}$ ​in ​the form $2^m\times 5^n$, ​where $m,n$ are non-negative integers. Hence, write its decimal expansion, without ​actual division.
   Show answer

   $5000=2^3\times 5^4$; hence $\frac{257}{5000}=\frac{514}{10^4}=0.0514$.

   Step-by-step solution →
4. Q44 Exemplar · Exercise 1.3
   Prove that $\sqrt{p}+\sqrt{q}$ is ​irrational, ​where $p$, $q$ ​are ​primes.
   Show answer

   By contradiction, assuming $\sqrt{p}+\sqrt{q}=r$ rational forces $\sqrt{p}=\frac{r^2+p-q}{2r}$ to be rational, contradicting that $\sqrt{p}$ is irrational. Hence $\sqrt{p}+\sqrt{q}$ is irrational.

   Step-by-step solution →
5. Q45 CBSE 2026
   The HCF of $960$ and $432$ is :

   1. (a)$48$
   2. (b)$54$
   3. (c)$72$
   4. (d)$36$
   Show answer

   (a) $48$

   Step-by-step solution →
6. Q46 CBSE 2026
   The LCM of $960$ and $240$ is :

   1. (a)$960$
   2. (b)$240$
   3. (c)$60$
   4. (d)$15$
   Show answer

   (a) $960$

   Step-by-step solution →
7. Q47 CBSE 2026
   $7\times 11\times 13+5$ is

   1. (a)a prime number.
   2. (b)an odd number.
   3. (c)a composite number.
   4. (d)a multiple of $5$.
   Show answer

   (c) a composite number.

   Step-by-step solution →
8. Q48 CBSE 2026
   The ​product ​of the H.C.F. ​and L.C.M. ​of two ​numbers $50$ and $20$ ​is:

   1. (a)$100$
   2. (b)$1000$
   3. (c)$50$
   4. (d)$20$
   Show answer

   (b) $1000$

   Step-by-step solution →
9. Q49 CBSE 2026
   If $\text{HCF}(850,325)$ is $25$, ​then $\text{LCM}(850,325)$ ​is:

   1. (a)$442$
   2. (b)$11050$
   3. (c)$8450$
   4. (d)$2210$
   Show answer

   (b) $11050$

   Step-by-step solution →
10. Q50 CBSE 2026
    The ​HCF ​of $2^2\times 3^3$ ​and $3^2\times 2^3$ is ​:

    1. (a)$1$
    2. (b)$2\times 3$
    3. (c)$2^2\times 3^2$
    4. (d)$2^3\times 3^3$
    Show answer

    (c) $2^2\times 3^2$

    Step-by-step solution →
11. Q51 CBSE 2026
    The HCF of ​the smallest ​prime number and the smallest 3-digit number is $2^m{5}^n$. The ​respective ​values of $m$ and $n$ are ​:

    1. (a)$0,0$
    2. (b)$1,0$
    3. (c)$0,1$
    4. (d)$1,1$
    Show answer

    (b) $1,0$

    Step-by-step solution →
12. Q52 CBSE 2026
    The ​HCF of $2^3$ ​and $3^2$ ​is :

    1. (a)$2^0\times 3^0$
    2. (b)$2^1\times 3^1$
    3. (c)$2^3\times 3^2$
    4. (d)$2^2\times 3^3$
    Show answer

    (a) $2^0\times 3^0$

    Step-by-step solution →
13. Q53 CBSE 2026
    The natural number $2$ is :

    1. (a)a prime number
    2. (b)a composite number
    3. (c)prime as well as composite
    4. (d)neither prime nor composite
    Show answer

    (a) a prime number

    Step-by-step solution →
14. Q54 CBSE 2026
    The natural number $1$ is :

    1. (a)A prime number.
    2. (b)A composite number.
    3. (c)Prime as well as composite.
    4. (d)Neither prime nor composite.
    Show answer

    (d) Neither prime nor composite.

    Step-by-step solution →
15. Q55 CBSE 2026
    Which of the following ​statements is true for HCF and LCM of two distinct natural numbers $a$ ​and $b$ ?

    1. (i)HCF ​is always greater ​than ​LCM.
    2. (ii)HCF is ​a factor of LCM.
    3. (iii)LCM ​is a ​factor of HCF.

    1. (a)(i) only
    2. (b)(i) and (iii)
    3. (c)(i) and (ii)
    4. (d)(ii) only
    Show answer

    (d) (ii) only

    Step-by-step solution →
16. Q56 CBSE 2026
    The ​LCM ​of ​two ​consecutive ​natural numbers $p$ ​and $p+1$ is :

    1. (a)$p$
    2. (b)$p^2+p$
    3. (c)$1$
    4. (d)$2p+1$
    Show answer

    (b) $p^2+p$

    Step-by-step solution →
17. Q57 CBSE 2026
    For any ​natural number $n$, $6^n$ ​ends with the ​digit ​:

    1. (a)$0$
    2. (b)$6$
    3. (c)$3$
    4. (d)$2$
    Show answer

    (b) $6$

    Step-by-step solution →
18. Q58 CBSE 2026
    For ​any ​natural number $n$, $5^n$ ​ends with the digit ​:

    1. (a)$0$
    2. (b)$5$
    3. (c)$3$
    4. (d)$2$
    Show answer

    (b) $5$

    Step-by-step solution →
19. Q59 CBSE 2026
    Which ​of the following ​numbers ​will not ​end ​with $0$ for ​any ​natural ​number $n$ ?

    1. (a)$4n$
    2. (b)$4^n$
    3. (c)$3^n+1$
    4. (d)$10^{n+1}$
    Show answer

    (b) $4^n$

    Step-by-step solution →
20. Q60 CBSE 2026
    $(3\times 11\times 13+3)$ is ​:

    1. (a)a prime number
    2. (b)divisible by $13$
    3. (c)a composite number
    4. (d)an odd number
    Show answer

    (c) a composite number

    Step-by-step solution →