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 16 of 17

1. Q301 CBSE 2020
   An army contingent of 612 ​members ​is to march behind an army band of 48 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?
   Show answer

   The maximum number of columns is $12$.

   Step-by-step solution →
2. Q302 CBSE 2020
   Using Euclid's ​algorithm, ​find the ​HCF of ​272 ​and ​1032.
   Show answer

   $\text{HCF}(272,1032)=8$.

   Step-by-step solution →
3. Q303 CBSE 2020
   Find HCF ​of $44$, $96$ ​and $404$ by prime factorization ​method. ​Hence find ​their LCM.
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   HCF $=4$ and LCM $=106656$.

   Step-by-step solution →
4. Q304 CBSE 2020
   Show that $(12{)}^n$ ​cannot end ​with ​digit $0$ or $5$ ​for any ​natural number $n$.
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   Since $12^n=2^{2n}\times 3^n$ has no factor of $5$, it is never divisible by $5$, so it cannot end in $0$ or $5$ for any $n\in \mathbb{N}$. Proved.

   Step-by-step solution →
5. Q305 CBSE 2020
   Show that the ​square of any ​positive ​integer cannot ​be ​of the form $(5q+2)$ or $(5q+3)$ for ​any integer $q$.
   Show answer

   A perfect square leaves remainder $0$, $1$ or $4$ on division by $5$, so it can never be of the form $5q+2$ or $5q+3$. Hence proved.

   Step-by-step solution →
6. Q306 CBSE 2020
   Prove ​that one ​of every three ​consecutive positive integers ​is divisible by $3$.
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   Writing $n=3q+r$ with $r\in \{0,1,2\}$: if $r=0$ then $n$, if $r=1$ then $n+2$, and if $r=2$ then $n+1$ is divisible by $3$. Hence one of every three consecutive positive integers is divisible by $3$. Q.E.D.

   Step-by-step solution →
7. Q307 CBSE 2020
   Prove ​that $(\sqrt{2}+\sqrt{5})$ is ​irrational.
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   The assumption that $\sqrt{2}+\sqrt{5}$ is rational leads to $\sqrt{2}$ being rational, a contradiction. Therefore $\sqrt{2}+\sqrt{5}$ is irrational. Proved.

   Step-by-step solution →
8. Q308 CBSE 2019
   If $\text{HCF}(336,54)=6$, ​find $\text{LCM}(336,54)$.
   Show answer

   $\text{LCM}(336,54)=3024$

   Step-by-step solution →
9. Q309 CBSE 2019
   The ​decimal expansion ​of $\frac{51}{2^3\times 5^2}$ will ​terminate ​after ​how ​many decimal ​places?
   Show answer

   The expansion terminates after $3$ decimal places ($0.255$).

   Step-by-step solution →
10. Q310 CBSE 2019
    The HCF ​of ​two numbers $a$ ​and $b$ ​is $5$ and their LCM is $200$. ​Find ​the ​product $ab$.
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    $ab=1000$.

    Step-by-step solution →
11. Q311 CBSE 2019
    Find ​after how many places ​of ​decimal ​the decimal form ​of ​the ​number $\frac{27}{2^3\times 5^4\times 3^2}$ will terminate.
    Show answer

    The decimal terminates after $4$ places (it equals $0.0006$).

    Step-by-step solution →
12. Q312 CBSE 2019
    Express $429$ as ​a ​product of its ​prime factors.
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    $429=3\times 11\times 13$

    Step-by-step solution →
13. Q313 CBSE 2019
    Find ​a rational ​number ​between $\sqrt{2}$ ​and $\sqrt{7}$.
    Show answer

    A rational number between $\sqrt{2}$ and $\sqrt{7}$ is $\frac{3}{2}=1.5$ (or any rational with $2<x^2<7$, e.g. $2$).

    Step-by-step solution →
14. Q314 CBSE 2019
    Write ​the ​number of zeroes ​in the ​end of a ​number whose prime ​factorization ​is $2^2\times 5^3\times 3^2\times 17$.
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    The number ends in $2$ zeroes.

    Step-by-step solution →
15. Q315 CBSE 2019
    Find ​a ​rational ​number between $\sqrt{2}$ ​and $\sqrt{3}$.
    Show answer

    A rational number between $\sqrt{2}$ and $\sqrt{3}$ is $1.5=\frac{3}{2}$ (other answers such as $1.6$ are also valid).

    Step-by-step solution →
16. Q316 CBSE 2019
    Write ​the ​smallest ​number ​which is ​divisible by ​both $306$ ​and $657$.
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    The smallest number divisible by both is $\text{LCM}(306,657)=22338$.

    Step-by-step solution →
17. Q317 CBSE 2019
    Find the ​HCF of $1260$ ​and $7344$ ​using Euclid's algorithm.
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    $\text{HCF}(1260,7344)=36$

    Step-by-step solution →
18. Q318 CBSE 2019
    Show ​that every ​positive odd ​integer is of the ​form $(4q+1)$ or $(4q+3)$, ​where $q$ ​is some integer.
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    Every positive odd integer is of the form $4q+1$ or $4q+3$ for some integer $q$. Proved.

    Step-by-step solution →
19. Q319 CBSE 2019
    The ​HCF of two ​numbers is $116$ ​and their ​LCM ​is $1740$. ​If one number is $580$, find ​the other.
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    The other number is $348$.

    Step-by-step solution →
20. Q320 CBSE 2019
    If ​HCF of $65$ and $117$ ​is ​expressible in the form $65n-117$, then ​find ​the ​value of $n$.
    Show answer

    $n=2$

    Step-by-step solution →