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Chapter 3 Playing With Numbers Ex. 3.2

Question 1.
What is the sum of any two:
(a) Odd numbers?
(b) Even numbers?
Solution:
(a) The sum of any two odd numbers is even.
(b) The sum of any two even numbers is even.

Question 2.
State whether the following statements are True or False.
(a) The sum of three odd numbers is even.
(b) The sum of two odd numbers and one even number is even.
(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 only the even prime number.
(i) All even numbers are composite numbers.
(j) The product of any two even numbers is always even.
Solution:
(а) False [∵ 3 + 5 + 7 = 15 (odd)]
(b) True [∵ 3 + 5 + 6 = 14 (even)]
(c) True [∵ 5 x 7 x 9 = 315 (odd)]
(d) False [∵ 6 + 2 = 3 (odd)]
(e) False [∵ 2 is a prime number but it is even]
(f) False [∵ 3 is a prime number having 1 and 3 as its factors]
(g) False [∵ 7 + 2 = 9 (odd)]
(h) True [∵ 2 is even and the lowest prime number]
(i) False [∵ 2 is even but not composite number]
(j) True [∵ 4 x 6 = 24 (even)]

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 up to 100.
Solution:
The required pair of prime numbers having same digits are:
(17 and 71), (37 and 73), (79 and 97).

Question 4.
Write down separately the prime and composite numbers less than 20.
Solution:
Prime numbers less than 20 are:
2, 3, 5, 7, 11, 13, 17 and 19
Composite numbers less than 20 are:
4, 6, 8, 9, 10, 12, 14, 15, 16 and 18

Question 5.
What is the greatest prime number between 1 and 10?
Solution:
The greatest prime number between 1 and 10 is 7.

Question 6.
Express the following as the sum of two odd primes.
(a) 44
(b) 36
(c) 24
(d) 18
Solution:
(a) 44 = 13 + 31
(b) 36 = 17 + 19
(c) 24 = 7 + 17
(d) 18 = 7 + 11

Question 7.
Give three pairs of prime numbers whose difference is 2.
[Remark: Two prime numbers whose difference is 2 are called twin primes]
Solution:
Required pairs are: (3 and 5), (5 and 7) and (11 and 13)

Question 8.
Which of the following numbers are prime?
(a) 23
(b) 51
(c) 37
(d) 26
Solution:
(a) 23 is a prime number [ ∵ 23 = 1 x 23]
(b) 51 is not a prime number [∵ 51 = 1 x 3 x 17]
(c) 37 is a prime number [∵ 37 = 1 x 37]
(d) 26 is not a prime number [∵ 26 = 1 x 2 x 13]

Question 9.
Write seven consecutive composite numbers less than 100 so that there is no prime number between them.
Solution:
Required seven consecutive composite numbers are:
90, 91, 92, 93, 94, 95 and 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:
(a) 21 can be expressed as 3 + 5 + 13
(b) 31 can be expressed as 5 + 7 + 19
(c) 53 can be expressed as 13 + 17 + 23
(d) 61 can be expressed as 11 + 13 + 37

Question 11.
Write five pairs of prime numbers less than 20 whose sum is divisible by 5.
(Hint: 3 + 7 = 10)
Solution:
Required pairs of prime numbers less than 20 are:
(i) 2 + 3 = 5
(ii) 2 + 13 = 15
(iii) 11 + 9 = 20
(iv) 17 + 3 = 20
(v) 7 + 13 = 20

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:
(a) prime number
(b) composite number
(c) prime, composite
(d) 2
(e) 4
(f) 2

Chapter 3 Playing With Numbers Ex. 3.2