RD Sharma Solutions: Playing With Numbers (Exercise 2.1)

# Playing With Numbers (Exercise 2.1) RD Sharma Solutions | Mathematics (Maths) Class 6 PDF Download

Q.1. Define

(i) factor

(ii) multiple

Give four examples of each.

Ans:

(i) Factor: A factor of a number is an exact divisor of that number.

For example, 4 exactly divides 32. Therefore, 4 is a factor of 32.

Examples of factors are:

2 and 3 are factors of 6 because 2 × 3 = 6

2 and 4 are factors of 8 because 2 × 4 = 8

3 and 4 are factors of 12 because 3 × 4 = 12

3 and 5 are factors of 15 because 3 × 5 = 15

(ii) Multiple: When a number 'a' is multiplied by another number 'b', the product is the multiple of both the numbers 'a' and 'b'.

Examples of multiples:

6 is a multiple of 2 because 2 × 3 = 6

8 is a multiple of 4 because 4 × 2 = 8

12 is a multiple of 6 because 6 × 2 = 12

21 is a multiple of 7 because 7 × 3 = 21

Q.2. Write all factors of each of the following numbers:

(i) 60

(ii) 76

(iii) 125

(iv) 729

Ans:

(i) 60 = 1 × 60

60 =  2 × 30

60 = 3 × 20

60 = 4 × 15

60 = 5 × 12

60 = 6 × 10

∴ The factors of 60 are 1, 2, 3, 4, 5, 6, 10, 12, 15, 20, 30 and 60.

(ii) 76 = 1 × 76

76 = 2 × 38

76 = 4 × 19

∴ The factors of 76 are 1, 2, 4, 19, 38 and 76.

(iii) 125 = 1 × 125

125 = 5 × 25

∴ The factors of 125 are 1, 5, 25 and 125.

(iv) 729 = 1 × 729

729 = 3 × 243

729 = 9 × 81

729 = 27 × 27

∴ The factors of 729 are 1, 3, 9, 27, 81, 243 and 729.

Q.3. Write first five multiples of each of the following numbers:

(i) 25

(ii) 35

(iii) 45

(iv) 40

Ans:

(i) The first five multiples of 25 are as follows:

25 × 1 = 25

25 × 2 = 50

25 × 3 = 75

25 × 4 = 100

25 × 5 = 125

(ii) The first five multiples of 35 are as follows:

35 × 1 = 35

35 × 2 = 70

35 × 3 = 105

35 × 4 = 140

35 × 5 = 175

(iii) The first five multiples of 45 are as follows:

45 × 1 = 45

45 × 2 = 90

45 × 3 = 135

45 × 4 = 180

45 × 5 = 225

(iv) The first five multiples of 40 are as follows:

40 × 1 = 40

40 × 2 = 80

40 × 3 = 120

40 × 4 = 160

40 × 5 = 200

Q.4. Which of the following numbers have 15 as their factor?

(i) 15615

(ii) 123015

Ans:

(i) 15 is a factor of 15,615 because it is a divisor of 15,615.

i.e., 1041 × 15 = 15,615

(ii) 15 is a factor of 1,23,015 because it is a divisor of 1,23,015.

i.e., 8,201 × 15 = 1,23,015

Thus, both the given numbers have 15 as their factor.

Disclaimer: The answer given in the book is incorrect.

Q.5. Which of the following numbers are divisible by 21?

(i) 21063

(ii) 20163

Ans: We know that a given number is divisible by 21 if it is divisible by each of its factors.

The factors of 21 are 1, 3, 7 and 21.

(i) Sum of the digits of the given number = 2 + 1 + 0 + 6 + 3 = 12 which is divisible by 3.

Hence, 21,063 is divisible by 3.

Again, a number is divisible by 7 if the difference between twice the one's digit and the number formed by the other digits is either 0 or a multiple of 7.

2,106 − (2 × 3) = 2,100 which is a multiple of 7.

Thus, 21,063 is divisible by 21.

(ii) Sum of the digits of the given number = 2 + 0 + 1 + 6 + 3 = 12 which is divisible by 3.

Hence, 20,163 is divisible by 3.

Again, a number is divisible by 7 if the difference between twice the one's digit and the number formed by the other digits is either 0 or multiple of 7.

2016 − (2 × 3) = 2010 which is not a multiple of 7.

Thus, 20,163 is not divisible by 21.

Q.6. Without actual division show that 11 is a factor of each of the following numbers:

(i) 1111

(ii) 11011

(iii) 110011

(iv) 1100011

Ans:

(i) 1,111

The sum of the digits at the odd places = 1 + 1 = 2

The sum of the digits at the even places = 1 + 1 = 2

The difference of the two sums =  2 − 2 = 0

∴ 1,111 is divisible by 11 because the difference of the sums is zero.

(ii) 11,011

The sum of the digits at the odd places = 1 + 0 + 1 = 2

The sum of the digits at the even places = 1 + 1 = 2

The difference of the two sums = 2 − 2 = 0

∴ 11,011 is divisible by 11 because the difference of the sums is zero.

(iii) 1,10,011

The sum of the digits at the odd places = 1 + 0 + 1 = 2

The sum of the digits at the even places = 1 + 0 + 1 = 2

The difference of the two sums = 2 − 2 = 0

∴ 1,10,011 is divisible by 11 because the difference of the sums is zero.

(iv) 11,00,011

The sum of the digits at the odd places = 1 + 0 + 0 + 1 = 2

The sum of the digits at the even places = 1 + 0 + 1 = 2

The difference of the two sums = 2 − 2 = 0

∴ 11,00,011 is divisible by 11 because the difference of the sums is zero.

Q.7. Without actual division show that each of the following numbers is divisible by 5:

(i) 55

(ii) 555

(iii) 5555

(iv) 50005

Ans: A number will be divisible by 5 if the unit's digit of that number is either 0 or 5.

(i) In 55, the unit's digit is 5. Hence, it is divisible by 5.

(ii) In 555, the unit's digit is 5. Hence, it is divisible by 5.

(iii) In 5,555, the unit's digit is 5. Hence, it is divisible by 5.

(iv) In 50,005, the unit's digit is 5. Hence, it is divisible by 5.

Q.8. Is there any natural number having no factor at all?

Ans: No, because each natural number is a factor of itself.

Q.9. Find numbers between 1 and 100 having exactly three factors.

Ans: The numbers between 1 and 100 having exactly three factors are 4, 9, 25, and 49.

The factors of 4 are 1, 2 and 4.

The factors of 9 are 1, 3 and 9.

The factors of 25 are 1, 5 and 25.

The factors of 49 are 1, 7 and 49.

Q.10. Sort out even and odd numbers:

(i) 42

(ii) 89

(iii) 144

(iv) 321

Ans:

A number which is exactly divisible by 2 is called an even number.

Therefore, 42 and 144 are even numbers.

A number which is not exactly divisible by 2 is called an odd number.

Therefore, 89 and 321 are odd numbers.

The document Playing With Numbers (Exercise 2.1) RD Sharma Solutions | Mathematics (Maths) Class 6 is a part of the Class 6 Course Mathematics (Maths) Class 6.
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## FAQs on Playing With Numbers (Exercise 2.1) RD Sharma Solutions - Mathematics (Maths) Class 6

 1. What is the concept of playing with numbers in mathematics?
Ans. Playing with numbers is a mathematical concept that involves manipulating numbers and performing various operations on them to solve problems and discover patterns. It includes topics like divisibility rules, factors, multiples, prime and composite numbers, and so on.
 2. How can playing with numbers help in improving mathematical skills?
Ans. Playing with numbers helps in improving mathematical skills by enhancing problem-solving abilities, logical thinking, and analytical reasoning. It also strengthens the understanding of number properties and operations, which are fundamental in mathematics.
 3. What are the divisibility rules in playing with numbers?
Ans. Divisibility rules are specific rules that determine whether a given number is divisible by another number without actually performing the division. For example, a number is divisible by 2 if its units digit is even, divisible by 3 if the sum of its digits is divisible by 3, and so on.
 4. How can prime and composite numbers be identified while playing with numbers?
Ans. Prime numbers are numbers that have exactly two distinct factors, 1 and the number itself. Composite numbers, on the other hand, have more than two factors. While playing with numbers, prime numbers can be identified by checking if they are divisible only by 1 and themselves, whereas composite numbers can have divisors other than 1 and the number itself.
 5. How can playing with numbers be applied in real-life situations?
Ans. Playing with numbers has practical applications in various real-life situations. For example, understanding divisibility rules can help in simplifying fractions or determining if a number is evenly divisible by another number. It can also aid in analyzing patterns and making predictions, which are essential in fields like finance, statistics, and computer science.

## Mathematics (Maths) Class 6

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