# NCERT Solutions for Class 10 Maths Chapter 1 Real Numbers (Ex 1.2) Exercise 1.2

The most important time to prepare for the challenge is now for all Class 10 students, who will take their first board exams, whether administered by the CBSE Board or any other state board. Board exams are the first nationalized exams for students in Class 10, which is the only difference. A student’s academic and professional prospects may advance if they pass the exams successfully. Additionally, a student’s performance on the board exam reflects their level of sincerity and sense of responsibility. Results from the Class 10 board serve as a determining factor in the admissions process. They serve as a crucial component of a resume and reflect a student’s academic standing. Many prestigious colleges and universities give students in Class 10 separate marks, determining which applicants will be admitted. As a result, Class 10 changes everything and shapes careers. Every subject is thoroughly studied, and students work to identify their areas of interest. They choose their stream based on that information and their exam results. This way, students’ decision-making skills improve post Class 10.

Exam stress and tension are overcome during the Class 10 board exams, which helps students become mentally strong and calm. Students will have the confidence to take and pass more difficult exams once they have completed and passed the Class 10 board exams. In order to perform well under pressure, students will learn how to manage their time and stress. Students taking their Class 10 board exams at different locations must step outside their comfort zone. Since the exam is created and graded by teachers rather than students, you have the opportunity to assess your performance more accurately. Students gain confidence for the future when they successfully navigate a challenging situation.

Class 10 topics are extensive, necessitating routine revision and practice. Additionally, learners may sometimes find it challenging to resolve exercise challenges. Extramarks presents the best study material, including NCERT solutions, PDF notes, and other materials, to assist students in thoroughly preparing for each exam section. Students can improve their exam strength using the NCERT Solutions Class 10 Maths Chapter 1 Exercise 1.2. The most recent CBSE syllabus is the foundation for the NCERT Solutions Class 10 Maths Chapter 1 Exercise 1.2 and Exercise 1.2 Class 10th is required to complete the exercise.

The chapter, the importance of the subject, and any subtopics must all be carefully read and understood by the students. It is necessary to comprehend the chapter and complete the final exam preparation fully. Additionally, before developing a strategy for final exam preparation, students need to familiarize themselves with the syllabus. The syllabus plays a significant role in achieving the highest possible exam grade. NCERT Solutions Class 10 Maths Chapter 1 Exercise 1.2, are available to students in Class 10 Mathematics in PDF format. As a result, using the Extramarks website and mobile application’s NCERT Solutions Class 10 Maths Chapter 1 Exercise 1.2 for students’ studies is a dependable and trustworthy study resource.

Most students do not create a proper study schedule to prepare for final exams. The main topic must receive enough time from the students. When studying for their exam, students should remember that the main exam will last three hours. Additionally, attempt the question with the higher score before moving on to the questions with 1 and 2 marks. Students can take exams to estimate how well they understand various subjects and topics. As a result, students must effectively prepare for exams, and it is advised that they regularly review a subject’s material. Every day, students are required to complete homework assignments and some papers from the previous school year. Furthermore, students can quickly understand the exam format and concept by completing sample papers and past year’s question papers.

The NCERT Solutions Class 10 Maths Chapter 1 Exercise 1.2 can also be used to self-assess ongoing Mathematics exam preparation. Students must therefore analyse their preparation for the Mathematics exam and review the Class 10 Maths Chapter 1 Exercise 1.2 Solutions. Students can easily access NCERT Solutions for Class 12 in all subjects through Extramarks. Additionally, NCERT Solutions Class 11 for all subjects are readily available to students in Class 11. Similarly, students looking for NCERT Solutions Class 10 for English can easily search for it on the Extramarks website. Each exercise at the end of the chapter has solutions in Extramarks Mathematics NCERT Solutions Class 9.

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**NCERT Solutions for Class 10 Maths Chapter 1 – Real Numbers **

The Real Numbers chapter begins with an introduction to real numbers in Mathematics, NCERT Class 10, Chapter 1. Real numbers were the subject of a brief discussion in ninth grade. In the first two sections, we delve even deeper into this subject by talking about two crucial real number properties:

- Euclid’s division algorithm
- The fundamental theorem of arithmetic

A brief explanation of Euclid’s division lemma makes up Euclid’s division algorithm. The following exercise is based on the subject of going over irrational numbers once more. Theorems are presented in various ways to help the student comprehend how a particular number can be shown to be irrational. Many examples are provided for the same. Exercise 1.2 is a brief exercise with only three questions. Moving on to the following exercise, which is based on a review of the topic of rational numbers and their decimal expansions. Questions in exercise 1.4 must be answered using three theorems. Students must therefore fully understand these theorems.

Students studying for the Class 10 Mathematics exam should have access to NCERT Solutions of Mathematics. Students can get all of their questions answered by using the resources provided by NCERT Solutions for Class 10 such as sample-solved questions, summaries, notes, and more. The best study resources, like the NCERT Solutions Class 10 Maths Chapter 1 Exercise 1.2, can be found by students by visiting the Extramarks website or mobile app. Moreover, students can better understand Chapter 1. Students should also review Exercise 1.2 from Chapter 1 of NCERT Solutions Class 10 Mathematics.

Students who need assistance with Exercise 1.2 in Class 10 can consult the NCERT Solutions Class 10 Maths Chapter 1 Exercise 1.2. All students have access to trustworthy and dependable educational materials through the learning platform Extramarks. Extramarks also make it simpler to find study materials. Students in Class 11 Central Board of Secondary Education can download the NCERT Solutions Class 10 Maths Chapter 1 Exercise 1.2 in PDF format from the Extramarks website and mobile application.

**Access NCERT Solutions for Class 10 Maths Chapter 1 – Real Numbers**

Some students are having trouble understanding how to answer the practice questions because Chapter 1 of Class 10 is so challenging. The Chapter 1 exercise in the NCERT books is excellent. NCERT Solutions Class 10 Maths Chapter 1 Exercise 1.2 can be used by students to assist them in completing the exercise. Students who have questions about the NCERT Solutions Class 10 Maths Chapter 1 Exercise 1.2 can ask them in real-time during live sessions on Extramarks’ website and mobile application.

The NCERT Solutions Class 10 Maths Chapter 1 Exercise 1.2, can be downloaded by students to help them study for their exams. On the Extramarks website and mobile app, students should visit the NCERT Solutions Class 10 Maths Chapter 1 Exercise 1.2.

As students get ready for their final exams, the NCERT Solutions Class 10 Maths Chapter 1 Exercise 1.2 encourage them to believe in themselves. Students can also quickly complete the Mathematics exercise and assess their responses to the NCERT Solutions Class 10 Maths Chapter 1 Exercise 1.2. Additionally, Class 10 students can comprehend the proper way to respond to Mathematics questions with the help of the NCERT Solutions Class 10 Maths Chapter 1 Exercise 1.2.

**Important Topics under NCERT Solutions for Class 10 Maths Chapter 1 Real Numbers (Ex 1.2) Exercise 1.2**

The goal of NCERT Solutions Class 10 Maths Chapter 1 Exercise 1.2 Real Numbers is to give students a solid learning resource. Students can learn the fundamentals covered in the chapter and can better understand and recognize the various questions in Class 10 Maths Chapter 1 Exercise 1.2 with the help of Extramarks solutions for NCERT Solutions Class 10 Maths Chapter 1 Exercise 1.2. The Class 10 Maths Chapter 1 Exercise 1.2 from Extramarks was created to help students learn about fundamental concepts and challenging real-world problems. Additionally, these NCERT Solutions Class 10 Maths Chapter 1 Exercise 1.2, support students in understanding the principles underlying the marking scheme according to the most recent syllabus.

The first chapter of the mathematics curriculum for class 10 is “Real Numbers.” It is an important mathematics chapter that is studied in class 10. To fully understand the chapter on real numbers, all four of its major sections must be read and comprehended.

The four significant subjects that make up the chapter Real Numbers are listed in the paragraphs that follow. For a clear understanding of the concepts, it is advised that students carefully go over these topics.

- Real Numbers: An Overview
- Arithmetic Fundamental Theorem (H.C.F. and L.C.M.)
- The Division Lemma of Euclid
- Return to Irrational Numbers

Students attending live lectures on a laptop, students sometimes become exhausted. Students can easily download the NCERT Solutions Class 10 Maths Chapter 1 Exercise 1.2 offline in PDF format thanks to Extramarks understanding of this. Students can practice easily and without having to sit in front of a laptop for an extended period by downloading the NCERT Solutions Class 10 Maths Chapter 1 Exercise 1.2.

The NCERT textbook contains a sizable number of questions that students can answer and practice. Students can improve their grades in their Class 10 exams by starting to practice with the NCERT Solutions Class 10 Maths Chapter 1 Exercise 1.2. NCERT Solutions Class 10 Maths Chapter 1 Exercise 1.2 is a good resource for students who are having trouble with Chapter 1. The Extramarks website and mobile app make it simple for students to access the NCERT Solutions Class 10 Maths Chapter 1 Exercise 1.2.

**Importance of NCERT Class 10 Maths Chapter 1 Real Numbers**

Real numbers are significant because they are used in practically every area of mathematics and everyday life. We advise students to get the most out of this chapter on real numbers to develop a strong number sense and be able to complete all the problems that real numbers-based solutions are required for in their exams.

Integers, decimals, and fractions are examples of rational numbers, whereas root overs, pi (22/7), and other irrational numbers are not. All numbers other than imaginary numbers are considered real numbers.

The NCERT Solutions Class 10 Maths Chapter 1 Exercise 1.2, are available on the Extramarks website and mobile application for students who need help with Chapter 1. Students in Class 10 can use the NCERT Solutions Class 10 Maths Chapter 1 Exercise 1.2, to verify their answers even after practising.

**NCERT Solutions for Class 10 Maths All Chapters**

There are 15 Chapters in NCERT Class 10 Mathematics books. Students can quickly check the NCERT Solutions for the Class 10 Mathematics all chapters.

**Real Numbers**

There are a total of four exercises in Real Numbers Class 10, each containing 18 problems.

“Prove Irrational,” “Problems based on Euclid’s division lemma,” “HCF and LCM,” and “Divisibility” were the most frequently asked topics on previous board exams. In addition, we cover the Fundamental Theorem of Arithmetic, significant positive integer properties, fraction to decimal conversions, and decimal to fraction conversions.

**Polynomials**

There are four exercises totaling 13 questions in the polynomials class 10 textbook. Scoring topics include issues with locating polynomials, properties zeros and coefficients, long division of polynomials, locating a quadratic polynomial, and locating polynomial zeros.

** Pair of Linear Equations in Two Variables**

There are seven exercises for Class 10’s pair of linear equations, totaling 55 problems. The solutions to these problems will simplify the solution of linear equations for you. The problems will be based on concepts like linear equations in two variables, algebraic methods for solving linear equations, the elimination method, the cross-multiplication method, time and work, age, boat stream, and being reducible to a pair of linear equations.

#### Quadratic Equations

There are four exercises in class 10 on quadratic equations, each of which has 24 problems. In board exams, questions about converting real-world problems into quadratic equations and finding the roots of quadratic equations tend to score highly.

**Arithmetic Progressions**

There are four exercises in Class 10 Arithmetic Progressions, each of which has 49 problems. Chapter 5 covers important topics such as finding the nth term and calculating the sum of n consecutive terms.

**Triangles**

64 problems in all six exercises for the class 10 assignment on triangles. The questions are based on triangle properties and nine important theorems for scoring high marks in CBSE Class 10 exams.

** Coordinate Geometry**

There are four exercises in class 10 coordinate geometry, each of which has 33 problems. Important examples in class 10 boards include questions about finding the distance between two points using their coordinates, the area of a triangle, and the line divided in the ratio (Section Formula).

**Introduction to Trigonometry**

There are a total of four exercises in Introduction to Trigonometry Class 10 that include 27 problems. The main topics your students learn about in this chapter are questions based on trigonometric ratios of particular angles, trigonometric identities, and trigonometric ratios of complementary angles. Trigonometry formulas are essential for achieving 100% in Board Exams.

**Some Applications of Trigonometry**

One exercise in some Applications of Trigonometry for Class 10 consists of 16 problems. In this chapter, students will learn about practical trigonometry applications, and the questions are based on those applications.

**Circles**

There are two exercises totaling 17 problems in Circle Class 10’s curriculum. Know terms like “tangent,” “secant,” “number tangents from a point to a circle,” and more.

**Constructions**

There are four exercises total, each of which contains 14 problems, for Constructions Class 10. Drawing tangents and making similar triangles are important topics covered in the questions.

**Areas Related to Circles**

Three exercises total 35 problems in the class 10 unit on Areas Related to Circles. Work out issues involving “Areas of Combinations of Plane Figures,” “Areas of a Circle’s Sector and Segment,” and “Areas of a Circle’s Perimeter and Area.”

**Surface Areas and Volumes**

Class 10 Surface Areas and Volumes has five exercises, each of which has 36 problems. The mensuration unit in CBSE Class 10 Mathematics includes the “Surface Areas and Volumes” chapter. Finding the areas and volumes of different solids, such as a cube, cuboid, cylinder, frustum, or combination of solids, is the basis for the problems.

**Statistics**

There are four exercises totaling 25 problems in Statistics Class 10’s curriculum. This chapter will explore issues with determining the mean, mode, or median of grouped data. Understand the concept of cumulative frequency distribution to answer questions.

**Class 10 Maths Chapter 1 – Exercise 1.2**

The HCF and LCM of given pairs, as well as the integers that allow users to find the product of any two numbers, are covered in the NCERT Solutions Class 10 Maths Chapter 1 Exercise 1.2, “Real Numbers.” Additionally, students should practice Exercise 1.2 from Chapter 1 of the NCERT Solutions for Class 10 Math. They explore the various features of numbers and how to use them to address problems in the study of numbers.

Students who use Extramarks’ analytical approach to Class 10 can fully prepare for their upcoming exams and comprehend the material. The answers to Exercises 1.1, 1.3, and 1.4 were all included in NCERT Solutions Class 10 Maths Chapter 1 Exercise 1.2.

Numerous problems and exercises are included in the NCERT textbook for students to complete. Students can improve their exam performance by practising with the NCERT Solutions Class 10 Maths Chapter 1 Exercise 1.2, for instance. NCERT Solutions Class 10 Maths Chapter 1 Exercise 1.2 is available for students who need help with Chapter 1. On the Extramarks website and mobile application, students can quickly download the NCERT Solutions Class 10 Maths Chapter 1 Exercise 1.2, and work out their solutions.

**NCERT Solutions for Class 10 Maths Chapter 1 All Exercises **

The students find Chapter 1 of Class 10 Mathematics to be challenging. It requires a lot of practice and understanding of the formula. There is a lot of pressure on students to perform well in their other classes as they begin their exam preparation. Students are currently seeking the best answers to their mathematics problems. One of the best platforms for education that helps students find solutions to their issues is Extramarks. Students can interact with the instructor during live lectures to get their questions about the subject answered.

Students can access the NCERT Solutions Class 10 Maths Chapter 1 Exercise 1.2, for instance, on the Extramarks website and mobile application if they are having issues with it. Even after practising, Class 10 students can use the NCERT Solutions Class 10 Maths Chapter 1 Exercise 1.2, to verify their responses.

Thanks to the NCERT Solutions for Class 10 Maths Chapter 1 Exercise 1.2, students gain confidence as they prepare for their final exams. Students can also quickly complete the mathematics review exercise and assess their performance on the NCERT Solutions Class 10 Maths Chapter 1 Exercise 1.2. Additionally, students in Class 10 can learn how to correctly respond to mathematics questions with the help of the NCERT Solutions Class 10 Maths Chapter 1 Exercise 1.2.

**Practical Problems from Maths Class 10 Chapter 1 Exercise 1.2**

There are many practical questions in Chapter 1 Class 10. The NCERT Solutions Class 10 Maths Chapter 1 Exercise 1.2 are helpful for students to enhance their understanding of the topic. Extramarks is one of the best learning platforms for study material. Students can also solve their doubts about practical questions from the NCERT Solutions Class 10 Maths Chapter 1 Exercise 1.2. Moreover, students can also download the NCERT Solutions Class 10 Maths Chapter 1 Exercise 1.2 PDF version offline.

**Did You Know?**

Any number that appears on the number line with the letter R typically qualifies as a real number. It can be an integer that is zero, negative, rational, irrational, or positive. Students can better understand the basic and complex ideas of real numbers by reading this chapter of Class 10 Mathematics Exercise 1.2.

**Q.1** Express each number as a product of its prime factors:

(i) 140 (ii) 156 (iii) 3825 (iv) 5005 (v) 7429

**Ans.**

$\begin{array}{l}\text{(i)}140=2\times 2\times 5\times 7={2}^{2}\times 5\times 7\\ \text{(ii)}156=2\times 2\times 3\times 13={2}^{2}\times 3\times 13\\ \text{(iii) 3825}=3\times 3\times 5\times 5\times 17={3}^{2}\times {5}^{2}\times 17\\ \text{(iv) 5005}=5\times 7\times 11\times 13\\ \text{(v) 7429}=17\times 19\times 23\end{array}$

**Q.2** Find the LCM and HCF of the following pairs of integers and verify that

LCM × HCF = product of the two numbers.

(i) 26 and 91

(ii) 510 and 92

(iii) 336 and 54

**Ans.**

$\begin{array}{l}\text{(i)}\\ 26=2\times 13\\ 91=7\times 13\\ \text{HCF of 26 and 91 = 13}\\ \text{and LCM of 26 and 91 = 2}\times 7\times 13=182\\ \text{Now,}\\ \text{26}\times 91=2\times 13\times 7\times 13\\ \text{\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em} \hspace{0.17em}\hspace{0.17em}\hspace{0.17em}}=(2\times 13\times 7)\times 13\\ \text{\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}}=\text{LCM}\times \text{HCF}\\ \text{(ii)}\\ 510=2\times 3\times 5\times 17\\ \text{\hspace{0.17em}\hspace{0.17em}}92=2\times 2\times 23\\ \text{HCF of}510\text{and 92 = 2}\\ \text{and LCM of}510\text{and 92 = 2}\times 2\times 3\times 5\times 17\times 23=23460\\ \text{Now,}\\ 510\times 92=2\times 3\times 5\times \; 17\times 2\times 2\times 23\\ \text{\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}}=(\text{2}\times 2\times 3\times 5\times 17\times 23)\times 2\\ \text{\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em} \hspace{0.17em}}=\text{LCM}\times \text{HCF}\\ \text{(iii)}\\ \text{336}=2\times 2\times 2\times 2\times 3\times 7={2}^{4}\times 3\times 7\\ \text{}54=2\times 3\times 3\times 3=2\times {3}^{3}\\ \text{HCF of 336 and 54}=2\times 3=6\\ \text{and}\\ \text{LCM of 336 and 54}={2}^{4}\times {3}^{3}\times 7=3024\\ \text{Now,}\\ \text{336}\times \text{54}={2}^{4}\times 3\times 7\times 2\times {3}^{3}\\ \text{\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}}=({2}^{4}\times {3}^{3}\times 7)\times 2\times 3\\ \text{\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}}=\text{LCM}\times \text{HCF}\end{array}$

**Q.3** Find the LCM and HCF of the following integers by applying the prime factorization method.

(i) 12, 15 and 21 (ii) 17, 23 and 29 (iii) 8, 9 and 25

**Ans.**

$\begin{array}{l}\text{(i)}\\ \text{12}={2}^{2}\times 3\\ 15=3\times 5\\ 21=3\times 7\\ \text{HCF}=3\\ \text{LCM}={2}^{2}\times 3\times 5\times 7=420\\ \\ \text{(ii)}\\ \text{17}=1\times 17\\ 23=1\times 23\\ 29=1\times 29\\ \text{HCF}=1\\ \text{LCM}=17\times 23\times 29=11,339\\ \\ \text{(iii)}\\ 8={2}^{3}\\ 9={3}^{2}\\ 25={5}^{2}\\ \text{HCF}=1\\ \text{LCM}={2}^{3}\times {3}^{2}\times {5}^{2}=1800\end{array}$

**Q.4** Given that HCF (306, 657) = 9, find LCM (306, 657)

**Ans.**

$\begin{array}{l}\text{We know that}\\ \text{HCF}\left(\text{a,\hspace{0.17em}b}\right)\text{\hspace{0.33em}\xd7\hspace{0.33em}LCM}\left(\text{a,\hspace{0.17em}b}\right)\text{= a\xd7b}\\ \Rightarrow \text{HCF}\left(\text{306, 657}\right)\text{\xd7 LCM}\left(\text{306, 657}\right)\text{= 306\xd7657}\\ \Rightarrow \text{LCM}\left(\text{306, 657}\right)\text{=}\frac{\text{306\xd7657}}{\text{HCF}\left(\text{306, 657}\right)}\text{}\\ \text{\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}=}\frac{\text{306\xd7657}}{\text{9}}\\ \text{\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}= 22338}\end{array}$

**Q.5** Check whether 6^{n} can end with the digit 0 for any natural number n.

**Ans.**

$\begin{array}{l}\text{A number ending with the digit 0 should be divisible by}\\ \text{10. It will also be divisible by 2 and 5 as}\\ \text{10 = 2 \xd7 5}\\ {\text{Prime factorisation of 6}}^{\text{n}}\text{=}{\left(\text{2 \xd73}\right)}^{\text{n}}\\ {\text{Here, we find that 5 is not a prime factor of 6}}^{\text{n}}\text{.}\\ {\text{Hence, for any value of n, 6}}^{\text{n}}\text{will not be divisible by 5.}\\ {\text{Therefore, 6}}^{\text{n}}\text{cannot end with the digit 0 for any}\\ \text{natural number n.}\end{array}$

**Q.6** Explain why 7 × 11 × 13 + 13 and 7 × 6 × 5 × 4 × 3 × 2 × 1 + 5 are composite number.

**Ans.**

$\begin{array}{l}7\times 11\times 13+13=13\times (7\times 11+1)\text{}\\ \text{}=\text{}13\times (77+1)\\ \text{}=\text{}13\times 78\text{}\\ \text{}=\text{}13\times 13\times 2\times 3\\ \text{The given expression is a composite number as it has}\\ \text{three factors 2, 3 and 13.}\\ \text{Now,}\\ 7\times 6\times 5\times 4\times 3\times 2\times 1+5=5\times (7\times 6\times 4\times 3\times 2\times 1+1)\\ \text{}=5\times (1008+1)\\ \text{}=5\times 1009\\ \text{The given expression is a composite number as it has}\\ \text{two factors other than 1 and the number itself.}\end{array}$

**Q.7** 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?

**Ans.**

Time taken by Sonia to complete 1 round of the field = 18 minutes

18 = 2 × 3 × 3

12 = 2 × 2 × 3

Therefore, Ravi and Sonia will meet together at the starting point

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