class 6th ncert math chapter 3rd

                                   Unnat Tuition Center

Chapter 3: Playing with Numbers - NCERT Solutions

Exercise 3.1

Question 1: Write all factors of the following numbers: (a) 24 (b) 15 (c) 21 (d) 27 (e) 12 (f) 20 (g) 18 (h) 23 (i) 36

Solution 1: Factors of a number are the exact divisors of that number. To find factors, we need to find all numbers that divide the given number exactly without leaving any remainder.

(a) 24 Factors of 24 are: 1, 2, 3, 4, 6, 8, 12, 24 (Because 24 is divisible by each of these numbers)

(b) 15 Factors of 15 are: 1, 3, 5, 15

(c) 21 Factors of 21 are: 1, 3, 7, 21

(d) 27 Factors of 27 are: 1, 3, 9, 27

(e) 12 Factors of 12 are: 1, 2, 3, 4, 6, 12

(f) 20 Factors of 20 are: 1, 2, 4, 5, 10, 20

(g) 18 Factors of 18 are: 1, 2, 3, 6, 9, 18

(h) 23 Factors of 23 are: 1, 23 (23 is a prime number, having only two factors: 1 and itself)

(i) 36 Factors of 36 are: 1, 2, 3, 4, 6, 9, 12, 18, 36

Question 2: Write first five multiples of: (a) 5 (b) 8 (c) 9

Solution 2: Multiples of a number are obtained by multiplying the number by natural numbers (1, 2, 3, 4, 5,...). We need to write the first five multiples.

(a) 5 First five multiples of 5 are: 5 × 1 = 5 5 × 2 = 10 5 × 3 = 15 5 × 4 = 20 5 × 5 = 25 So, the first five multiples of 5 are: 5, 10, 15, 20, 25

(b) 8 First five multiples of 8 are: 8 × 1 = 8 8 × 2 = 16 8 × 3 = 24 8 × 4 = 32 8 × 5 = 40 So, the first five multiples of 8 are: 8, 16, 24, 32, 40

(c) 9 First five multiples of 9 are: 9 × 1 = 9 9 × 2 = 18 9 × 3 = 27 9 × 4 = 36 9 × 5 = 45 So, the first five multiples of 9 are: 9, 18, 27, 36, 45

Question 3: Match the items in Column 1 with the items in Column 2:

(Column 1) (Column 2) (i) 35 (a) Multiple of 8 (ii) 15 (b) Multiple of 7 (iii) 16 (c) Multiple of 70 (iv) 20 (d) Factor of 30 (v) 25 (e) Factor of 50 (f) Factor of 20 (g) Factor of 18

Solution 3: Let's match each item in Column 1 with the correct description in Column 2.

(i) 35 35 is a multiple of 7 (35 = 7 × 5). Match: (i) - (b) Multiple of 7

(ii) 15 15 is a factor of 30 (30 ÷ 15 = 2). Match: (ii) - (d) Factor of 30

(iii) 16 16 is a multiple of 8 (16 = 8 × 2). Match: (iii) - (a) Multiple of 8

(iv) 20 20 is a factor of 20 (20 ÷ 20 = 1) and also a factor of 50 is not correct. Factor of 20 is more appropriate. Match: (iv) - (f) Factor of 20  

(v) 25 25 is a factor of 50 (50 ÷ 25 = 2) and also a factor of 25 (25 ÷ 25 = 1). Factor of 50 is more appropriate from the options. Match: (v) - (e) Factor of 50  

Final Matches:

• (i) - (b)
• (ii) - (d)
• (iii) - (a)
• (iv) - (f)
• (v) - (e)

Question 4: Find all the multiples of 9 upto 100.

Solution 4: We need to list the multiples of 9 that are less than or equal to 100. We can find them by multiplying 9 by natural numbers sequentially until the product exceeds 100.

9 × 1 = 9 9 × 2 = 18 9 × 3 = 27 9 × 4 = 36 9 × 5 = 45 9 × 6 = 54 9 × 7 = 63 9 × 8 = 72 9 × 9 = 81 9 × 10 = 90 9 × 11 = 99 9 × 12 = 108 (This is greater than 100, so we stop here)

Therefore, the multiples of 9 upto 100 are: 9, 18, 27, 36, 45, 54, 63, 72, 81, 90, 99.

---

Exercise 3.2

Question 1: What is the sum of any two (a) Odd numbers? (b) Even numbers?

Solution 1:

(a) Sum of any two Odd numbers: Let's take a few examples of odd numbers and find their sum:

• 1 + 3 = 4 (Even)
• 5 + 7 = 12 (Even)
• 9 + 11 = 20 (Even) In all cases, the sum of two odd numbers is an even number.

(b) Sum of any two Even numbers: Let's take a few examples of even numbers and find their sum:

• 2 + 4 = 6 (Even)
• 6 + 8 = 14 (Even)
• 10 + 12 = 22 (Even) In all cases, the sum of two even numbers is an even number.

Answer: (a) Even number (b) Even number

Question 2: State whether the following statements are true (T) or false (F): (a) The sum of three odd numbers is even. (b) The sum of two even numbers and one odd number is odd. (c) The product of three odd numbers is odd. (d) If an even number is divided by 2, the quotient is always odd. (e) All prime numbers are odd. (f) Prime numbers do not have any factors. (g) Sum of two prime numbers is always even. (h) 2 is the only even prime number. (i) All even numbers are composite numbers. (j) The product of two even numbers is always even.  

Solution 2: Let's analyze each statement:

(a) The sum of three odd numbers is even. (F)

• False. Odd + Odd = Even, and then Even + Odd = Odd. So, the sum of three odd numbers is always odd. Example: 1 + 3 + 5 = 9 (Odd).

(b) The sum of two even numbers and one odd number is odd. (F)

• False. Even + Even = Even, and then Even + Odd = Odd. So, the sum of two even numbers and one odd number is always odd. Example: 2 + 4 + 3 = 9 (Odd). Correction: Statement is True. Sum of two even numbers is EVEN. Even + Odd = Odd. So the statement should be TRUE.
• Corrected (b) True (T) - My initial check was incorrect.

(c) The product of three odd numbers is odd. (T)

• True. Odd × Odd = Odd, and then Odd × Odd = Odd. Example: 1 × 3 × 5 = 15 (Odd).  

(d) If an even number is divided by 2, the quotient is always odd. (F)

• False. For example, 4 is even, and 4 ÷ 2 = 2, which is even, not odd.

(e) All prime numbers are odd. (F)

• False. The number 2 is a prime number, but it is even. All other prime numbers (except 2) are odd.

(f) Prime numbers do not have any factors. (F)

• False. Prime numbers have exactly two factors: 1 and themselves. So, they do have factors.

(g) Sum of two prime numbers is always even. (F)

• False. If we add two prime numbers, and one of them is 2 (the only even prime), and the other is an odd prime, their sum will be odd. Example: 2 + 3 = 5 (Odd). However, if we add two odd prime numbers, their sum is even (Odd + Odd = Even).

(h) 2 is the only even prime number. (T)

• True. By definition, all other even numbers are divisible by 2 and also by other numbers, so they cannot be prime (except 2 itself).

(i) All even numbers are composite numbers. (F)

• False. The number 2 is an even number, but it is a prime number, not a composite number. Composite numbers are even numbers greater than 2 and all even numbers except 2 are composite.

(j) The product of two even numbers is always even. (T)

• True. Any even number is divisible by 2. So, if you multiply two even numbers, the product will also be divisible by 2, and hence even.

Corrected Truth Values: (a) False (F) (b) True (T) (c) True (T) (d) False (F) (e) False (F) (f) False (F) (g) False (F) (h) True (T) (i) False (F) (j) True (T)

Question 3: The numbers 13 and 31 are prime numbers. Both these numbers have same digits 1 and 3. Find such pairs of prime numbers upto 100.

Solution 3: We need to find pairs of prime numbers up to 100 where the digits are reversed to get another prime number. Let's check prime numbers and their digit reversals:

• 13 and 31 (given in the question) - Both are prime.
• 17 and 71 - Both are prime.
• 37 and 73 - Both are prime.
• 79 and 97 - Both are prime.

Let's check other primes and their reversals to ensure we haven't missed any: Primes less than 100: 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97.

Reversals (and check if prime):

• 19 reversed is 91 (not prime, 91 = 7 × 13)
• 23 reversed is 32 (even, not prime except 2)
• 29 reversed is 92 (even, not prime except 2)
• 41 reversed is 14 (even, not prime except 2)  
• 43 reversed is 34 (even, not prime except 2)
• 47 reversed is 74 (even, not prime except 2)
• 53 reversed is 35 (not prime, 35 = 5 × 7)
• 59 reversed is 95 (not prime, 95 = 5 × 19)
• 61 reversed is 16 (even, not prime except 2)
• 67 reversed is 76 (even, not prime except 2)
• 83 reversed is 38 (even, not prime except 2)
• 89 reversed is 98 (even, not prime except 2)

So, the pairs of prime numbers with reversed digits (upto 100) are: (13, 31), (17, 71), (37, 73), (79, 97).

Question 4: Write down separately the prime and composite numbers less than 20.

Solution 4: Numbers less than 20 are: 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19.

• Prime numbers are numbers greater than 1 that have only two factors: 1 and themselves. Prime numbers less than 20 are: 2, 3, 5, 7, 11, 13, 17, 19

• Composite numbers are numbers greater than 1 that have more than two factors. Composite numbers less than 20 are: 4, 6, 8, 9, 10, 12, 14, 15, 16, 18

Note: The number 1 is neither prime nor composite.

Question 5: What is the greatest prime number between 1 and 10?

Solution 5: Prime numbers between 1 and 10 are: 2, 3, 5, 7. Among these prime numbers, the greatest is 7.

Answer: 7

Question 6: Express the following as the sum of two odd primes: (a) 44 (b) 36 (c) 24 (d) 18

Solution 6: We need to express each given even number as a sum of two odd prime numbers. (Goldbach's Conjecture states that every even integer greater than 2 can be expressed as the sum of two primes - we're finding odd prime sums here).

(a) 44 We need to find two odd prime numbers that add up to 44. Let's try starting with the smallest odd prime, 3. 44 - 3 = 41. Is 41 a prime number? Yes. So, 44 = 3 + 41 (Both 3 and 41 are odd primes) Another possible answer: 44 = 13 + 31 (Both 13 and 31 are odd primes). There might be other possibilities too.

(b) 36 36 - 5 = 31. Is 31 a prime number? Yes. So, 36 = 5 + 31 (Both 5 and 31 are odd primes) Another possible answer: 36 = 7 + 29 (Both 7 and 29 are odd primes).

(c) 24 24 - 5 = 19. Is 19 a prime number? Yes. So, 24 = 5 + 19 (Both 5 and 19 are odd primes) Another possible answer: 24 = 7 + 17 (Both 7 and 17 are odd primes).

(d) 18 18 - 5 = 13. Is 13 a prime number? Yes. So, 18 = 5 + 13 (Both 5 and 13 are odd primes) Another possible answer: 18 = 7 + 11 (Both 7 and 11 are odd primes).

Answers (one possibility for each): (a) 44 = 3 + 41 (b) 36 = 5 + 31 (c) 24 = 5 + 19 (d) 18 = 5 + 13

Question 7: Give three pairs of prime numbers whose difference is 2.

Solution 7: Pairs of prime numbers whose difference is 2 are called twin primes. We need to find three such pairs.

Let's check prime numbers and look for pairs with a difference of 2:

• 3 and 5 (5 - 3 = 2). Both 3 and 5 are prime. So, (3, 5) is a twin prime pair.
• 5 and 7 (7 - 5 = 2). Both 5 and 7 are prime. So, (5, 7) is a twin prime pair.
• 11 and 13 (13 - 11 = 2). Both 11 and 13 are prime. So, (11, 13) is a twin prime pair.

Three pairs of prime numbers whose difference is 2 are: (3, 5), (5, 7), (11, 13).

Question 8: Which of the following numbers are prime? (a) 23 (b) 51 (c) 37 (d) 26

Solution 8: To determine if a number is prime, we need to check if it has factors other than 1 and itself.

(a) 23: Let's check for factors of 23. We only need to check divisibility by prime numbers up to √23 (which is roughly 4.8). Prime numbers less than 4.8 are 2, 3.

• 23 is not divisible by 2 (not even).
• 23 is not divisible by 3 (2+3=5, not divisible by 3). Since 23 is not divisible by 2 or 3, and we've checked all primes up to its square root, 23 is a prime number.

(b) 51: Let's check for factors of 51. Sum of digits 5 + 1 = 6, which is divisible by 3. So, 51 is divisible by 3. 51 ÷ 3 = 17. So, 51 = 3 × 17. Since 51 has factors other than 1 and itself (3 and 17), 51 is a composite number, not prime.

(c) 37: Let's check for factors of 37. We need to check divisibility by primes up to √37 (roughly 6). Primes less than 6 are 2, 3, 5.

• 37 is not divisible by 2 (not even).
• 37 is not divisible by 3 (3+7=10, not divisible by 3).
• 37 is not divisible by 5 (does not end in 0 or 5). Since 37 is not divisible by 2, 3, or 5, and we've checked all primes up to its square root, 37 is a prime number.

(d) 26: 26 is an even number, so it is divisible by 2. 26 = 2 × 13. Since 26 has factors other than 1 and itself (2 and 13), 26 is a composite number, not prime.

Answer: The prime numbers are (a) 23 and (c) 37.

Question 9: Write seven consecutive composite numbers less than 100 so that there is no prime number between them.

Solution 9: We need to find a sequence of 7 consecutive composite numbers below 100, meaning no prime number should exist within this sequence. Let's look for gaps between prime numbers.

Consider numbers around 90. Prime numbers near 90 are ... 83, 89, 97, ... After 89, the next prime number is 97. Let's look at numbers between 89 and 97. Numbers after 89 are 90, 91, 92, 93, 94, 95, 96, 97... Let's check if 90, 91, 92, 93, 94, 95, 96 are all composite.

• 90 - divisible by 2, 3, 5, 6, 9, 10, etc. (Composite)
• 91 - 91 = 7 × 13 (Composite)
• 92 - divisible by 2 (Composite)
• 93 - 93 = 3 × 31 (Composite)
• 94 - divisible by 2 (Composite)
• 95 - divisible by 5 (Composite)
• 96 - divisible by 2, 3, etc. (Composite)  
• 97 - is a prime number.

So, the seven consecutive composite numbers are: 90, 91, 92, 93, 94, 95, 96. There is no prime number between 90 and 96 (or including 90 and 96).

Answer: 90, 91, 92, 93, 94, 95, 96

Question 10: Express each of the following numbers as the sum of three odd primes: (a) 21 (b) 31 (c) 53 (d) 61

Solution 10: We need to express each number as a sum of three odd prime numbers. Let's try to use small odd primes like 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 41, 43, ...

(a) 21 Let's start with the smallest odd prime, 3. 21 - 3 = 18. Now we need to express 18 as a sum of two odd primes. We found earlier (Q6(d)) that 18 = 5 + 13 (or 7 + 11). Let's use 7 + 11. So, 21 = 3 + 7 + 11 (3, 7, 11 are all odd primes).

(b) 31 Let's start with 3 and 5 (smallest odd primes). 3 + 5 = 8. 31 - 8 = 23. Is 23 a prime number? Yes. So, 31 = 3 + 5 + 23 (3, 5, 23 are all odd primes). Another possibility: 31 = 5 + 7 + 19 (5, 7, 19 are all odd primes).

(c) 53 Let's start with 3, 7 (small odd primes). 3 + 7 = 10. 53 - 10 = 43. Is 43 a prime number? Yes. So, 53 = 3 + 7 + 43 (3, 7, 43 are all odd primes). Another possibility: 53 = 13 + 17 + 23 (13, 17, 23 are all odd primes).  

(d) 61 Let's try 3, 7, 11 (small odd primes). 3 + 7 + 11 = 21. 61 - 21 = 40. 40 is not prime. Let's try different combinations. Let's try 7 and 13. 7 + 13 = 20. 61 - 20 = 41. Is 41 prime? Yes. So, 61 = 7 + 13 + 41 (7, 13, 41 are all odd primes). Another possibility: 61 = 19 + 29 + 13 (19, 29, 13 are all odd primes).

Answers (one possibility for each): (a) 21 = 3 + 7 + 11 (b) 31 = 3 + 5 + 23 (c) 53 = 3 + 7 + 43 (d) 61 = 7 + 13 + 41

Question 11: Write five pairs of prime numbers less than 20 whose sum is divisible by 5.

Solution 11: We need to find pairs of prime numbers (less than 20) such that their sum is divisible by 5. For a sum to be divisible by 5, the sum must end in 0 or 5.

Let's list prime numbers less than 20: 2, 3, 5, 7, 11, 13, 17, 19.

Let's check pairs and their sums:

• 2 + 3 = 5 (Divisible by 5) - Pair: (2, 3)
• 2 + 5 = 7 (Not divisible by 5)
• 2 + 7 = 9 (Not divisible by 5)
• 2 + 13 = 15 (Divisible by 5) - Pair: (2, 13)
• 2 + 17 = 19 (Not divisible by 5)
• 3 + 2 = 5 (Already counted as (2, 3))
• 3 + 7 = 10 (Divisible by 5) - Pair: (3, 7)
• 3 + 17 = 20 (Divisible by 5) - Pair: (3, 17)
• 5 + 5 = 10 (Divisible by 5) - Pair: (5, 5) (Although 5 is repeated, it's still a pair of primes and their sum is divisible by 5).
• 7 + 3 = 10 (Already counted as (3, 7))
• 7 + 13 = 20 (Divisible by 5) - Pair: (7, 13)
• 11 + 9 = 20 (9 is not prime)
• 13 + 2 = 15 (Already counted as (2, 13))
• 17 + 3 = 20 (Already counted as (3, 17))
• 19 + 1 = 20 (1 is not prime)

Five pairs of prime numbers less than 20 whose sum is divisible by 5 are: (2, 3), (2, 13), (3, 7), (3, 17), (7, 13) and also (5, 5) can be included, but the question asks for pairs, typically implying distinct numbers. Let's use the first five distinct pairs we found.

Answer: (2, 3), (2, 13), (3, 7), (3, 17), (7, 13)

Question 12: Fill in the blanks: (a) A number which has only two factors is called a _______. (b) A number which has more than two factors is called a _______. (c) 1 is neither _______ nor _______. (d) The smallest prime number is _______. (e) The smallest composite number is _______. (f) The smallest even number is _______.

Solution 12: Let's fill in the blanks with the correct terms based on definitions:

(a) A number which has only two factors is called aprime number**.** (b) A number which has more than two factors is called acomposite number**.** (c) 1 is neitherprime** nor composite.** (d) The smallest prime number is2**.** (e) The smallest composite number is4**.** (f) The smallest even number is2**.**