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 10 of 17

1. Q181 CBSE 2024
   Find ​the smallest ​number that ​is divisible by each of 8, 9 ​and ​10.
   Show answer

   The smallest number divisible by each of $8$, $9$ and $10$ is $360$.

   Step-by-step solution →
2. Q182 CBSE 2024
   Can the number $(15{)}^n$, $n$ being ​a ​natural number, ​end ​with ​the digit $0$ ​? ​Give reasons.
   Show answer

   No. $(15{)}^n=3^n\times 5^n$, and by the Fundamental Theorem of Arithmetic this factorisation is unique, so $(15{)}^n$ has no factor $2$. Hence it is not divisible by $10$ and cannot end with the digit $0$ for any natural number $n$.

   Step-by-step solution →
3. Q183 CBSE 2024
   Given that ​HCF $(306,1314)=18$, ​find LCM ​of $(306,1314)$.
   Show answer

   LCM $(306,1314)=22338$

   Step-by-step solution →
4. Q184 CBSE 2024
   Find ​the ​HCF of $84$ and $144$ ​by the ​prime factorisation ​method.
   Show answer

   HCF$(84,144)=2^2\times 3=12$

   Step-by-step solution →
5. Q185 CBSE 2024
   Three bells toll at intervals ​of $9$, $12$ ​and $15$ minutes respectively. ​If ​they start tolling together, after what time will they ​next ​toll together ​?
   Show answer

   They will next toll together after $180$ minutes, that is, after $3$ hours.

   Step-by-step solution →
6. Q186 CBSE 2024
   Prove that $7-3\sqrt{5}$ ​is an ​irrational ​number, given that $\sqrt{5}$ is ​an ​irrational number.
   Show answer

   $7-3\sqrt{5}$ is irrational. Assuming $7-3\sqrt{5}=\frac{a}{b}$ gives $\sqrt{5}=\frac{7b-a}{3b}$, a rational number, which contradicts the fact that $\sqrt{5}$ is irrational. Hence proved.

   Step-by-step solution →
7. Q187 CBSE 2024
   Prove that $6-4\sqrt{5}$ is an irrational ​number, given ​that $\sqrt{5}$ ​is ​an ​irrational number.
   Show answer

   Assuming $6-4\sqrt{5}=\frac{a}{b}$ (integers $a$, $b$, $b\ne 0$) gives $\sqrt{5}=\frac{6b-a}{4b}$, a rational number, which contradicts the given fact that $\sqrt{5}$ is irrational. Hence $6-4\sqrt{5}$ is irrational. Hence proved.

   Step-by-step solution →
8. Q188 CBSE 2024
   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: assuming $-7-2\sqrt{3}=\frac{a}{b}$ (integers $a$, $b$, $b\ne 0$) gives $\sqrt{3}=-\frac{a+7b}{2b}$, a rational number, which contradicts the given fact that $\sqrt{3}$ is irrational. Hence proved.

   Step-by-step solution →
9. Q189 CBSE 2024
   Show ​that $11\times 19\times 23+3\times 11$ ​is not a ​prime ​number.
   Show answer

   $11\times 19\times 23+3\times 11=11\times (437+3)=11\times 440=4840$. It has the factor $11$, which is neither $1$ nor the number itself, so it is a composite number and not a prime number.

   Step-by-step solution →
10. Q190 CBSE 2024
    Explain why $(7\times 11\times 13+2\times 11)$ ​is not ​a ​prime ​number.
    Show answer

    $7\times 11\times 13+2\times 11=11\times (7\times 13+2)=11\times 93=1023$. Since $11$ is a factor of it other than $1$ and itself (in fact $1023=3\times 11\times 31$), it is a composite number, not a prime.

    Step-by-step solution →
11. Q191 CBSE 2024
    Find ​the ​greatest ​number which ​divides $65$ and $117$ ​completely.
    Show answer

    The greatest number which divides $65$ and $117$ completely is $13$.

    Step-by-step solution →
12. Q192 CBSE 2024
    Using prime factorization, ​find the HCF ​and ​LCM ​of $6,72$ ​and $120$.
    Show answer

    $\text{HCF}=6$ and $\text{LCM}=360$

    Step-by-step solution →
13. Q193 CBSE 2024
    Find whether ​each of ​the following ​is ​an irrational ​number ​or a ​rational ​number.

    1. (i)$(\sqrt{5}-\sqrt{3}{)}^2$
    2. (ii)$(5+\sqrt{3})(5-\sqrt{3})$
    Show answer

    (i) $(\sqrt{5}-\sqrt{3}{)}^2=8-2\sqrt{15}$, which is an **irrational** number; (ii) $(5+\sqrt{3})(5-\sqrt{3})=22$, which is a **rational** number.

    Step-by-step solution →
14. Q194 CBSE 2024
    Prove that $5-2\sqrt{3}$ is an irrational number. ​It ​is ​given ​that $\sqrt{3}$ ​is ​an irrational number.
    Show answer

    Assuming $5-2\sqrt{3}=\frac{a}{b}$ gives $\sqrt{3}=\frac{5b-a}{2b}$, a rational number, which contradicts the given fact that $\sqrt{3}$ is irrational. Hence $5-2\sqrt{3}$ is irrational.

    Step-by-step solution →
15. Q195 CBSE 2024
    There ​is a circular path around a sports field. Three cyclists start from the same ​point and at the ​same time and go in the ​same direction. If they take $30$ minutes, $40$ minutes ​and $48$ minutes respectively to complete ​one round ​of the field, ​after how many minutes will they meet again at the starting point ?
    Show answer

    They meet again at the starting point after $240$ minutes ($4$ hours).

    Step-by-step solution →
16. Q196 CBSE 2024
    Find ​the ​smallest 4-digit number ​exactly divisible by ​15, ​24 and ​36.
    Show answer

    The smallest 4-digit number exactly divisible by $15$, $24$ and $36$ is $1080$.

    Step-by-step solution →
17. Q197 CBSE 2024
    Show that the number $5\times 11\times 17+3\times 11$ ​is ​a ​composite ​number.
    Show answer

    $5\times 11\times 17+3\times 11=11\times (5\times 17+3)=11\times 88=2^3\times 11^2=968$. It has the factor $11$, which is neither $1$ nor the number itself, so it is a composite number.

    Step-by-step solution →
18. Q198 CBSE 2024
    Find ​LCM ​and ​HCF of two ​numbers $336$ and $54$, using ​prime-factorisation ​method.
    Show answer

    HCF $=6$ and LCM $=3024$.

    Step-by-step solution →
19. Q199 CBSE 2024
    Two ​alarm clocks ring their alarms at ​regular intervals of $20$ minutes ​and $25$ ​minutes respectively. If they first beep together at $12$ noon, at what time ​will ​they beep again together ​next time?
    Show answer

    The two alarms beep together again $100$ minutes after $12$ noon, that is, at $1:40$ p.m.

    Step-by-step solution →
20. Q200 CBSE 2024
    Prove ​that $6+3\sqrt{2}$ ​is ​an irrational number, given that $\sqrt{2}$ is ​an ​irrational ​number.
    Show answer

    $6+3\sqrt{2}$ is irrational. Assuming it equals $\frac{a}{b}$ gives $\sqrt{2}=\frac{a-6b}{3b}$, a rational number, which contradicts the irrationality of $\sqrt{2}$. Hence proved.

    Step-by-step solution →