NCERT Solutions for Class 6 Maths Chapter 3: Playing with Numbers - Exercise 3.1

Exercise 3.1

1. Write all the factors of the following numbers:

1. 24

Ans: \[24 = 1 \times 24 = 2 \times 12 = 3 \times 8 = 4 \times 6 = 6 \times 4\]

$\therefore $ Factors of 24 = 1, 2, 3, 4, 6, 12, 24.

2. 15

Ans: \[15 = 1 \times 15 = 3 \times 5 = 5 \times 3\]

$\therefore $ Factors of 15 = 1, 3, 5, 15.

3. 21

Ans: \[21 = 1 \times 21 = 3 \times 7 = 7 \times 3\]

$\therefore $ Factors of 21 = 1, 3, 7, 21.

4. 27

Ans: \[27 = 1 \times 27 = 3 \times 9 = 9 \times 3\]

$\therefore $ Factors of 27 = 1, 3, 9, 27.

5. 12

Ans: \[12 = 1 \times 12 = 2 \times 6 = 3 \times 4 = 4 \times 3\]

$\therefore $ Factors of 12 = 1, 2, 3, 4, 6, 12.

6. 20

Ans: \[20 = 1 \times 20 = 2 \times 10 = 4 \times 5 = 5 \times 4\]

$\therefore $ Factors of 20 = 1, 2, 4, 5, 10, 20.

7. 18

Ans: \[18 = 1 \times 18 = 2 \times 9 = 3 \times 6\]

$\therefore $ Factors of 18 = 1, 2, 3, 6, 9, 18.

8. 23

Ans: \[23 = 1 \times 23\]

$\therefore $ Factors of 23 = 1, 23.

9. 36

Ans: \[36 = 1 \times 36 = 2 \times 18 = 3 \times 12 = 4 \times 9 = 6 \times 6\]

$\therefore $ Factors of 36 = 1, 2, 3, 4, 6, 9, 12, 18, 36.

2. Write first five multiples of:

1. 5

Ans: \[5 \times 1 = 5,{\text{ }}5 \times 2 = 10,{\text{ }}5 \times 3 = 15,{\text{ }}5 \times 4 = 20,{\text{ }}5 \times 5 = 25\]

$\therefore $ First five multiples of 5 are 5, 10, 15, 20, 25.

2. 8

Ans: \[8 \times 1 = 8,{\text{ }}8 \times 2 = 16,{\text{ }}8 \times 3 = 24,{\text{ }}8 \times 4 = 32,{\text{ }}8 \times 5 = 40\]

$\therefore $ First five multiples of 8 are 8, 16, 24, 32, 40.

3. 9

Ans: \[9 \times 1 = 9,{\text{ }}9 \times 2 = 18,{\text{ }}9 \times 3 = 27,{\text{ }}9 \times 4 = 36,{\text{ }}9 \times 5 = 45\]

$\therefore $ First five multiples of 9 are 9, 18, 27, 36, 45.

3. Match the items in column 1 with the items in column 2:

Ans: 

1. $35 = 5 \times 7$

2. $15 = \dfrac{{30}}{2}$

3. $16 = 8 \times 2$

4. $20 = 20 \times 1$

5. $25 = \dfrac{{50}}{2}$

4. Find all the multiples of 9 up to 100.

Ans: Multiples of 9 up to 100 are: 9, 18, 27, 36, 45, 54, 63, 72, 81, 90, 99.

Class 6 Maths exercise 3.1

Mathematics is intricately combined with everyday activity of our lives. The entire concept of modern trade and commerce is dependent upon principles of mathematics. Mathematics cannot be done without; on this basis, the students should figuratively consider mathematics as the centre of all the culture, for it influences all aspects of a community. 

Mathematics influences art as well; for example, numerical formulas can be discovered in the musical harmonies. It also enriches every student who studies the subject. It helps the students improve various skills such as analytical thinking, helps with the power of reasoning. It also trains the mind in making quick decisions. These combined enable the students to be more curious about the things they are surrounded, thus helping them with better learning. 

Every aspect of our modern life is dependent on some form of technology. We are no longer simple nomadic farmers, but modern humans involved in a variety of projects which we further use to enhance the community. Technology has been an essential part of the reasons for the advancement of community and advanced technology is not possible without the use of mathematics. The technology will only advance in the future, becoming even more significant than life itself and hence so the Maths will continue to grow as well.

That is why students having strong fundamentals in mathematics will help them and their country by excelling in all crucial aspects of life. To help the students, the chapter contains the following topics:

1. Introduction

2. Factors and multiples

3. Exercise 3.1 Questions and Answers

4. Prime and composite numbers

5. Exercise 3.2 Questions and Answers

6. Tests for Divisibility of Numbers 

7. Exercise 3.3 Questions and Answers

8. Common Factors and Common Multiples

9. Exercise 3.4 Questions and Answers

10. Some more Divisibility Rules

11. Prime Factorization

12. Exercise 3.5 Questions and Answers

13. Highest Common Factor 

14. Exercise 3.6 Questions and Answers

15. Lowest Common Multiple

16. Some Problems on HCF and LCM 

After reading through the chapter, students are suggested to solve the exercises presented at the end of the section to evaluate their understanding of these concepts of great importance. Solving these exercises also help the students prepare better for the examinations and get excellent results.

Introduction

This segment of the chapter teaches the students about the importance of numbers. Through various examples and activities, this segment tries to make the student learn about the various characteristics and features of the numbers. It makes the student understand how a particular number can be written in many different ways. The students also understand how a specific number can be written as a product of two numbers. Through the activity mentioned in this segment of the chapter, the chapter makes the student understand how to figure out various direct divisors of the number. Once these are followed, the students have explained the proper definition of the term called ‘factors.’ The student is encouraged to explore the meaning of the factors further via a suggested activity.

Factors and Multiples

This chapter helps the students to understand valuable lessons about the fundamentals of numbers. The students learn how to find out a list of numbers which might be precisely divisible with a particular number. The chapter takes the student through various examples to show how a specific number can be written as divisors of other numbers. Through this activity, the section gives a clear idea to the students about the meaning of Factors. Through an elaborate game and activity, the chapter tries to explain to the students how a number is a multiple of each of its factors. Once this gets established, the segment moves further to teach some facts which might be interesting about the concept of factors and its multiples.

Through the elaborate exercise and experiments, the students learn if there can be a number, which is a common factor for all possible numbers and what number would that be. After finding that number, which is standard across all numbers, the chapter helps the students to find out about the probability of every number, whether it can be a factor to itself or not. This understanding helps in getting a better idea of the subject. After getting a clear picture of the above concepts, the segment moves on to convey to the students if every factor is also an exact divisor of that number or not. Through examples that this chapter showcases, this vital knowledge about the nature of the elements is conveyed to the students.

The students might start to wonder whether there exists only a particular number of factors of a specific number, or they can be infinite. Once the students understand these various intricate details about the character of the numbers, the chapter urges the students to move even further to understand other essential features, such as if all the multiple is higher than that number or is there a chance of any number being smaller. The students also get an idea of a multiple of a number can be equal or the same in value to that mentioned number.

The chapter containing exercise 3.1 Class 6 doesn’t stop here. The students can also learn other valuable things if they carry on reading. When the students read forward, they get the idea about what is the nature of multiples and whether there are given several multiples of a particular number or are they infinite. One establishing these ideas, the chapter establishes another fundamental idea about the nature of the multiple. The student learns whether a number can be a multiple of itself or not.

Through this crucial information, the segment enables the student to understand the complicated concept of what is called a perfect number and how it is defined. The students also learn how the sum of all the factors influences the meaning of the whole number.

Exercise 3.1 Questions and Answers

This exercise at the very start of the chapter is essential for you to test whether the student will be able to understand further lessons of mathematics properly or not. The exercise starts by asking the students to find out the factors of all the given numbers. With the second question, the exercise about the students’ ability to take out multiples of a particularly given number. The exercise doesn’t stop there; in order to enhance the students learning further, the exercise tries to test the student through various tricks and puzzles. The next question the student is asked to solve by matching the right answers to the given questions. This is to test the students’ understanding of whether they can still process the principles of the lesson taught if the question is presented to the students in more complex forms than what is usually presented.

Prime and Composite Numbers

Once the chapter Ex 3.1 Class 6 Maths makes it clear to the students about the concepts about a factor of a number and its various characteristics and the chapter moves on the segment of and teaches students about different other things. The students learn whether 1 has many factors or not; with that understanding, the students are informed about the concept of some number just having two factors only. The students get to know what such numbers are called and what two numbers can only be used as a factor of such a number.

This section further encourages the students to try to find out more prime numbers so that they get a better grip on the concept. Once that is cleared, the chapter uses this concept to teach the students about the concepts of a composite number. The students get to know how many factors are required to be defined as a composite number.

The section prepares the student of an excellent method by the famous mathematician from Greece Eratosthenes and teaches them how using this method the students could find a prime number without checking to calculate the factors of that particular number. The students are also informed about the critical dates of these events.

The section teaches students how to use the concepts of primes numbers and then identify them. And how the great mathematician Eratosthenes applied that concept to quickly calculate through its multiples the numbers which were not prime and what is the exact process of Steve of Eratosthenes which the students can use for their advantage.

The section next makes the students learn about the very fundamentals of the types of numbers in mathematics. After reading this, the students come to know what are the concepts of the odd number and what are the concepts of the even numbers. For example, what can be the smaller number, which number is an even number and what can be the lowest number, and so on. The students also learn about the only number, which is a prime number and also an even number.

Exercise 3.2 Questions and Answers

This exercise, which is next after the students are taught about the vital principle of a prime number, and composite, further helps the students to make their grip stronger on the subject. The exercise starts straight away with testing the students on their understanding of odd and even numbers by asking them to do various sums using them. The students are mostly used to the very straight cut questions, often which doesn’t allow them to think about the question or apply their brains to solve questions that seem trickier. So This exercise tests the students in formats such as true or false, to help the students find the right answers even in times when all answers seem right. The exercise brings to notice the uniqueness of numbers like 13 and 31. After the students get the concept about what makes these numbers unique, the exercise further urges the students to discover more such numbers. 

The exercise tests the knowledge of the students about how well they understand the prime numbers and composite numbers and asks them to write down a number of them. Using the same idea, the students are urged to write the greatest prime numbers up to 10 in the exercise. Prime numbers cannot be understood by getting the knowledge of how to add them, which is the question of the exercise. The students are required to add two prime numbers before moving to the next question of twin primes. Twin primes and the problems are asked to the students for solving it based on their understanding of prime numbers, before moving on to test the skills of the students about composite numbers. Furthermore, to test the acumen of the students a set of fill in the blanks is also presented to the students in the exercise.

Tests for Divisibility of Numbers 

This part of the chapter makes the student understand the notions about the laws of a number being divisible. This part of the section, through several activities and examples, explains the students about particular numbers and its divisibility. First, the number 10 is taken, and many of its multiples are looked at. This part teaches the students how to identify numbers that are divisible by 10 and also how to be quick in the process of their identification.

This part of the chapter moves on to teach the students about the numbers which are divisible by 5. However, through various observations, the student can learn to identify quickly the numbers which can be divided by 5 and how to locate numbers which are precisely divisible and which number may leave a remainder.

This part of the chapter is about the number which is divisible by the only even prime number, which is 2. To help the students, the section teaches them how to identify the numbers which have a probability of being divided by 2 and hot to do so as quickly as possible. The chapter also informs the students that what nature of the last digit is to be present in a given number helps in the determination of it being divisible by 2.

Divisibility by number 3 is also an essential part of the chapter which students learn about. This part takes various numbers to present to student’s different situations and in those situations are tested what numbers have a pattern and in spite of the pattern what number can be divisible and cannot be divided by 3 and how divisible are the multiples of 3 by 3.

This segment of the Class 6 Maths Chapter 3.1 gives the next level understanding, required for further progress, of the nature of the divisibility of the numbers. The students are taught about the characteristics of the number 6 and what numbers are itself divisible. Then these individual numbers are taken into account, and the properties of their compatibility are cross-checked. Further, based on this compatibility, the students learn what can be concluded whether any number which is divisible by these numbers can also be divided by 6 or not.

The next segment of this chapter, under the topic of divisibility of numbers, will focus on the numbers that can be divided by number 4. Again the students have presented a group of numbers and asked to study the patterns in the numbers. Through this pattern, the students learn about how to observe numbers that have three or more than three digits and what principle can be used to understand if the number will be divisible by 4 by calculating the last two digits of that number.

A series of numbers are presented to the students, and they are asked to observe in great detail each digit and their placement and the numbers as a whole and try to understand and see if the students manage to find patterns in them. For further studies, the number that is taken is number 8. After carefully searching and trying to locate patterns in a series of numbers, the students learn the relation between the digits of a number that has 4 or more digits. They will also learn about a number that is formed by calculating the sum of its last three digits and how to use these to quickly decide whether the number is divisible by 8 or not.

Divisibility by the number 9 is also taken into account in this segment of the chapter. A series of numbers which have more than 4 digits are mostly taken into this case study, and their patterns are broken down into simple formats to understand and use this understanding in various ways. One of the vital information the students learn whether or not the sum of all the digits of the number is divisible by the number, which is 9, then the whole number is also divided by the number 9 or not.

How numbers are divisible by the number 11 is next tested in the chapter. The students learn about some characteristics of the number 11. Many numbers having a variety of digits are taken for this experiment. Through this experiment, the student can learn about how he can use the various arrangements of the digits to calculate if the whole number will be divisible by 11 or not.

Exercise 3.3 Questions and Answers

Divisibility of numbers is an important characteristic of numbers which all students who are studying mathematics. The exercise starts with the students being given a variety of numbers, and the students are asked if they can be divided by another given set of numbers and to make a chart of the entire results. The students are encouraged to deploy divisibility tests for the next series of questions and are provided with a number of digits for the students to find out how various numbers can be used to divide them. In order to strengthen the students’ concept about the principle of divisibility, the exercise tries to ask a trick question by asking the students to complete the incomplete answers.

Common Factors and Common Multiples

Through ex 3.1 Class 6 students get to learn about other important aspects of the numbers. A pair of various numbers are taken for an activity to enable the student to understand these concepts better. Several factors are further deduced from those numbers. Through these deductions, the students get to understand the concepts about what is called co-prime numbers, and then the various examples of such pairs are given for the students. Then the students are asked to carry forward the activity with multiple other numbers. Common factors and divisors are calculated of these numbers. 

Exercise 3.4 Questions and Answers

The students at the very onset of the exercise are asked to solve some common factors of the given numbers. Once the students are successful in finding out the factors of the given numbers and they are able to do successfully, then the exercise moves on to test the students’ skills about their ability to find out multiples of numbers. Only after the lessons about multiples are excelled by the students, then the exercise moves on to tests the students’ ability to find common multiples of given numbers.

After asking students questions about numbers, which were co-prime, the exercise presents certain questions where two factors of a number are given. Using this the students are tested if they can find out other factors and if a given number is divisible by a particular number, the students are tested if they can find out what other numbers can also be divisible.

Some More Divisibility Rules

More rules about the process of divisibility are explored. To leave no stone unturned, the students are informed about the remaining rules, which will help them to improve on the subject. Some other principles the students get to learn are whether a number which is also divisible by another, then does all the factors can also be used to divide it or not.

The students learn further about what happens when a particular number is divisible by two co-prime numbers. The students also get to know whether this number can also be divided by their products. 

The students may also wonder what happens if two numbers which are presented are divisible by a particular number and how that number can also be divided by the sum of the given numbers. Similarly, the students also get the knowledge about how two given numbers react when divided by a particular number and how the difference of that number is also divisible by that number.

Prime Factorisation

The students who are reading this part of the chapter will understand various things regarding the concepts of factors. The students will get to know when a number is factorized and what is meant by a number as the product of its factors. Using these concepts, this part makes the student understand the definitions of prime factorizations and what happens if all the factors of a number are prime numbers.

Exercise 3.5 Questions and Answers

The exercise starts with a series of truths and false activity to test the students’ ability to figure out the right and wrong principles of these concepts. This exercise also provides factor trees in questions presented to the students and are asked to write down the numbers which are missing. This exercise helps the students to understand if they know which factors are included and not included in the prime factorization of a composite number. The students are asked if they can write the greatest number of 4 digits and also manage to express it in a manner of its prime factors. Similarly, the students are then asked in the exercise to write the smallest 5-digit number and express it in its prime factors. The exercise further asks the students to write down the prime numbers in ascending order in a particular question. The students are presented with a number of expressions of various numbers and are then asked about their status of prime factorization. 

Furthermore, in the exercise, certain numbers are provided to the students to test their ability to divide numbers. The students are encouraged to find out which is the number having four completely different prime factors, which is also the smallest.

Highest Common Factor 

To understand these concepts, students are encouraged to for an activity to take a few numbers and calculate the common factors of these. After taking out the common factors the students are taught how to take out the highest common factors between the two numbers. The students can also get to earn whether the highest common factor of two numbers or more numbers is also the highest of all the numbers, which form their common factors. Through these understandings, the student also gets knowledge about the concept of the Greatest Common Divisor.

Exercise 3.6 Questions and Answers

This exercise gives the students a variety of numbers and asks the students to calculate their HCF. Then the exercise asks the students to find out the HCF under particular conditions further. Further, on the exercise, HCF of a particular number and inquires if the numbers are correct or not.

Lowest Common Multiple

The students are asked about a common multiple of some numbers, and they are then asked to calculate the number which is the smallest but is also a factor of both numbers. Through this, the students understand the notion about what is also called the lowest common multiple is also the lowest of their common multiple or not. This section further teaches students other methods that they can deploy to calculate LCM.

Some Problems on HCF and LCM 

The students may often get confused between the two concepts of HCF and LCM. To clear all confusion and further improve the understanding of the concepts, the students are given various sets of problems to solve. This is very crucial because the students will not be able to understand the next chapter without having strong fundamentals about these.

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