Class 10 Maths · Chapter 1

Real Numbers

336 Class 10 Maths questions on Real Numbers from the NCERT textbook, NCERT Exemplar and CBSE board papers 2018–2026, each with its answer and a step-by-step solution.

CBSE Class 10 Maths previous year papers, with answers →

Questions · page 13 of 17

1. Q241 CBSE 2023
   Find ​the ​HCF ​and ​LCM of 72 and ​120.
   Show answer

   HCF $=24$, LCM $=360$

   Step-by-step solution →
2. Q242 CBSE 2023
   Show ​that $8^n$ can ​never ​end ​with the ​digit $0$ ​for any natural number $n$.
   Show answer

   Since $8^n=2^{3n}$ has no factor 5, it is never divisible by 10, so it can never end in 0.

   Step-by-step solution →
3. Q243 CBSE 2023
   Find ​the HCF ​of the ​numbers $540$ and $630$, using ​prime ​factorization ​method.
   Show answer

   $\text{HCF}(540,630)=90$

   Step-by-step solution →
4. Q244 CBSE 2023
   Find LCM ​and ​HCF of $96$ ​and $160$, ​using prime ​factorisation method.
   Show answer

   HCF $=32$ and LCM $=480$.

   Step-by-step solution →
5. Q245 CBSE 2023
   Show ​that $(15{)}^n$ cannot end ​with the ​digit $0$ ​for ​any natural ​number ​'n'.
   Show answer

   $(15{)}^n=3^n\times 5^n$ has no factor of $2$, so it is never divisible by $10$ and cannot end with $0$. Hence proved.

   Step-by-step solution →
6. Q246 CBSE 2023
   Find the ​LCM and HCF ​of $92$ ​and $510$, using ​prime ​factorisation.
   Show answer

   HCF $=2$, LCM $=23460$

   Step-by-step solution →
7. Q247 CBSE 2023
   Using ​prime factorisation, ​find ​HCF ​and ​LCM ​of $96$ and $120$.
   Show answer

   HCF $=24$, LCM $=480$

   Step-by-step solution →
8. Q248 CBSE 2023
   Find the ​greatest number ​which ​divides $85$ and $72$ ​leaving ​remainders $1$ ​and $2$ respectively.
   Show answer

   The greatest such number is $14$.

   Step-by-step solution →
9. Q249 CBSE 2023
   Prove ​that $\sqrt{3}$ ​is an ​irrational ​number.
   Show answer

   $\sqrt{3}$ is irrational (proved by contradiction).

   Step-by-step solution →
10. Q250 CBSE 2023
    Prove ​that $3+2\sqrt{2}$ ​is ​an ​irrational number, given that $\sqrt{2}$ is ​irrational.
    Show answer

    $3+2\sqrt{2}$ is irrational (proved by contradiction).

    Step-by-step solution →
11. Q251 CBSE 2023
    Show ​that $5+2\sqrt{3}$ is an irrational number, given ​that $\sqrt{3}$ ​is ​an irrational ​number.
    Show answer

    $5+2\sqrt{3}$ is irrational. Hence proved.

    Step-by-step solution →
12. Q252 CBSE 2023
    Find by ​prime ​factorisation ​the LCM ​of ​the numbers $18180$ ​and $7575$. Also, find the HCF of the two ​numbers.
    Show answer

    HCF $=1515$ and LCM $=90900$.

    Step-by-step solution →
13. Q253 CBSE 2023
    The traffic ​lights ​at three ​different road crossings change ​after every $48$ ​seconds, $72$ seconds and $108$ seconds respectively. ​If they ​change simultaneously at $7$ a.m., at what time will they change together next ?
    Show answer

    $7$ : $07$ : $12$ a.m. (after 432 s = 7 min 12 s)

    Step-by-step solution →
14. Q254 CBSE 2023
    Three bells ring ​at intervals of $6$, $12$ ​and $18$ ​minutes. ​If all the three ​bells rang at $6$ ​a.m., when will ​they ring together again?
    Show answer

    They ring together again at $6:36$ a.m.

    Step-by-step solution →
15. Q255 CBSE 2023
    Prove that $(7-2\sqrt{3})$ is ​an ​irrational number, given that $\sqrt{3}$ ​is an ​irrational number.
    Show answer

    $7-2\sqrt{3}$ is irrational (proved by contradiction).

    Step-by-step solution →
16. Q256 CBSE 2023
    Prove ​that $4+2\sqrt{3}$ is an ​irrational number, given that $\sqrt{3}$ ​is ​an ​irrational ​number.
    Show answer

    The assumption that $4+2\sqrt{3}$ is rational forces $\sqrt{3}$ to be rational, a contradiction. Hence $4+2\sqrt{3}$ is irrational.

    Step-by-step solution →
17. Q257 CBSE 2023
    Find ​the HCF ​and ​LCM of 26, 65 ​and ​117, using ​prime ​factorisation.
    Show answer

    HCF $=13$, LCM $=1170$

    Step-by-step solution →
18. Q258 CBSE 2023
    Prove that $\sqrt{2}$ ​is ​an irrational ​number.
    Show answer

    The assumption leads to a contradiction, so $\sqrt{2}$ is irrational.

    Step-by-step solution →
19. Q259 CBSE 2023
    Prove ​that $2+3\sqrt{3}$ is ​an irrational ​number. It is given that $\sqrt{3}$ is an ​irrational ​number.
    Show answer

    The assumption leads to $\sqrt{3}$ being rational, a contradiction; hence $2+3\sqrt{3}$ is irrational.

    Step-by-step solution →
20. Q260 CBSE 2023
    Prove that $3-2\sqrt{5}$ is ​an ​irrational ​number, ​given that $\sqrt{5}$ ​is an irrational ​number.
    Show answer

    $3-2\sqrt{5}$ is irrational (proved by contradiction).

    Step-by-step solution →