Chapter 1 · Real Numbers

NCERT Exemplar Class 10 Maths Exercise 1.3

All 14 questions of NCERT Exemplar Class 10 Maths Exercise 1.3 (Real Numbers), in the book's order, each with its answer and a step-by-step solution.

1. Q1
   Show that ​the square of ​any positive ​integer ​is ​either of the ​form $4q$ ​or $4q+1$ for some integer $q$.
   Show answer

   If $n$ is even, $n^2=4m^2=4q$; if $n$ is odd, $n^2=4(m^2+m)+1=4q+1$. So every square is of the form $4q$ or $4q+1$. Proved.

   Step-by-step solution →
2. Q2
   Show that ​cube of any positive ​integer ​is ​of ​the form $4m$, $4m+1$ or $4m+3$, for some integer $m$.
   Show answer

   For every positive integer $n$, the cube $n^3$ is of the form $4m$, $4m+1$ or $4m+3$. Hence proved.

   Step-by-step solution →
3. Q3
   Show that the ​square ​of any positive ​integer cannot ​be of ​the ​form $5q+2$ or $5q+3$ ​for any ​integer $q$.
   Show answer

   Every square of a positive integer is of the form $5q$, $5q+1$ or $5q+4$, hence never $5q+2$ or $5q+3$. Proved.

   Step-by-step solution →
4. Q4
   Show ​that ​the square ​of any positive integer cannot be ​of the form $6m+2$ ​or $6m+5$ ​for ​any ​integer $m$.
   Show answer

   A perfect square mod 6 can only be $0,1,3$ or $4$, never $2$ or $5$; hence no square has the form $6m+2$ or $6m+5$.

   Step-by-step solution →
5. Q5
   Show ​that the square of ​any ​odd ​integer is of ​the ​form $4q+1$, ​for some integer $q$.
   Show answer

   For any odd $n=2m+1$, $n^2=4(m^2+m)+1=4q+1$ with $q=m^2+m$ an integer.

   Step-by-step solution →
6. Q6
   If $n$ ​is an ​odd integer, ​then ​show that $n^2-1$ is ​divisible by ​8.
   Show answer

   With $n=2k+1$, $n^2-1=4k(k+1)$; since $k(k+1)$ is even, $n^2-1=8m$, so it is divisible by 8.

   Step-by-step solution →
7. Q7
   Prove that ​if $x$ and $y$ ​are both odd ​positive integers, ​then $x^2+y^2$ ​is even but not ​divisible ​by ​4.
   Show answer

   Proved: $x^2+y^2=4(m^2+m+n^2+n)+2$, which is even (a multiple of 2) but leaves remainder 2 on division by 4.

   Step-by-step solution →
8. Q8
   Use Euclid's ​division ​algorithm ​to ​find ​the ​HCF of $441,567,693$.
   Show answer

   The HCF of $441,567,693$ is $63$.

   Step-by-step solution →
9. Q9
   Using Euclid's division algorithm, find the ​largest ​number that divides 1251, 9377 and 15628 ​leaving remainders 1, 2 and ​3, ​respectively.
   Show answer

   The largest such number is $625$.

   Step-by-step solution →
10. Q10
    Prove ​that $\sqrt{3}+\sqrt{5}$ is ​irrational.
    Show answer

    By contradiction, assuming $\sqrt{3}+\sqrt{5}=r$ rational forces $\sqrt{3}=\frac{r^2-2}{2r}$ to be rational, which is false. Hence $\sqrt{3}+\sqrt{5}$ is irrational.

    Step-by-step solution →
11. Q11
    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 →
12. Q12
    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 →
13. Q13
    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 →
14. Q14
    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 →