Chapter 1 · Real Numbers

NCERT Class 10 Maths Exercise 1.1

All 7 questions of NCERT Class 10 Maths Exercise 1.1 (Real Numbers), in the book's order, each with its answer and a step-by-step solution.

1. Q1
   Express each ​number as a product of ​its ​prime factors: ​

   1. (i)$140$ ​
   2. (ii)$156$
   3. (iii)$3825$
   4. (iv)$5005$ ​
   5. (v)$7429$
   Show answer

   $140=2^2\times 5\times 7$; $156=2^2\times 3\times 13$; $3825=3^2\times 5^2\times 17$; $5005=5\times 7\times 11\times 13$; $7429=17\times 19\times 23$.

   Step-by-step solution →
2. Q2
   Find the LCM ​and HCF of ​the following pairs of integers and verify that $\text{LCM}\times \text{HCF}=$ ​product ​of ​the two numbers.

   1. (i)$26$ and $91$ ​
   2. (ii)$510$ and $92$
   3. (iii)$336$ and $54$
   Show answer

   (i) HCF $=13$, LCM $=182$; (ii) HCF $=2$, LCM $=23460$; (iii) HCF $=6$, LCM $=3024$. In each case $\text{HCF}\times \text{LCM}$ equals the product of the two numbers.

   Step-by-step solution →
3. Q3
   Find the ​LCM and HCF of the following ​integers ​by applying ​the prime factorisation method.

   1. (i)$12$, $15$ ​and $21$ ​
   2. (ii)$17$, $23$ and $29$ ​
   3. (iii)$8$, $9$ and $25$
   Show answer

   (i) HCF $=3$, LCM $=420$; (ii) HCF $=1$, LCM $=11339$; (iii) HCF $=1$, LCM $=1800$.

   Step-by-step solution →
4. Q4
   Given ​that ​HCF (306, ​657) = ​9, find ​LCM ​(306, ​657).
   Show answer

   $\text{LCM}(306,657)=22338$.

   Step-by-step solution →
5. Q5
   Check whether $6^n$ can end with ​the digit $0$ ​for ​any natural ​number $n$.
   Show answer

   No. $6^n=2^n\times 3^n$ has no factor of $5$, so it is never a multiple of $10$ and cannot end in $0$ for any natural number $n$. Its unit digit is always $6$.

   Step-by-step solution →
6. Q6
   Explain ​why $7\times 11\times 13+13$ and $7\times 6\times 5\times 4\times 3\times 2\times 1+5$ are ​composite ​numbers.
   Show answer

   $7\times 11\times 13+13=13\times 78=1014$ and $7\times 6\times 5\times 4\times 3\times 2\times 1+5=5\times 1009=5045$. Each is a product of two whole numbers greater than $1$, so both are composite.

   Step-by-step solution →
7. Q7
   There ​is a circular path around a sports ​field. Sonia takes 18 minutes to drive one round of the field, while Ravi takes 12 minutes ​for the same. Suppose ​they both start at the same point and at the same time, and go in ​the same direction. After how many minutes will they meet again at ​the starting point?
   Show answer

   They will meet again at the starting point after $36$ minutes.

   Step-by-step solution →