**Playing with Numbers**

**Exercise 3.2**

**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.

**Example – **1+5 = 6 , 7+9 = 16

(b) The sum of any two even numbers is even.

**Example – **4+2** = **6 , 8+4 = 12

**Ex 3.2 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)]

**Ex 3.2 Question 3.**

**The numbers 13 and 31 are prime numbers. Both these numbers have the same digits 1 and 3. Find such pairs of prime numbers up to 100.**

**Solution:**

The required pair of prime numbers having the same digits are:

(17 and 71), (37 and 73), (79 and 97).

**Ex 3.2 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

**Ex 3.2 Question 5.**

**What is the greatest prime number between 1 and 10?**

**Solution:**

The greatest prime number between 1 and 10 is 7.

**Ex 3.2 Question 6.**

**Express the following as the sum of two odd primes.**

**(a) 44**

**(b) 36**

**(c) 24**

**(d) 18**

**Solution:**

(a) 27 + 17 = 44

(b) 5 + 31 = 36

(c) 19 +5 = 24

(d) 15 + 3 = 18

**Ex 3.2 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:**

(3 and 5), (5 and 7) and (11 and 13)

**Ex 3.2 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]

**Ex 3.2 Question 9.**

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

**Solution:**

Seven consecutive composite numbers are:

90, 91, 92, 93, 94, 95 and 96

**Ex 3.2 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 11 + 7+ 3

(b) 31 can be expressed as 3 + 11 + 17

(c) 53 can be expressed as 23 + 17 + 13

(d) 61 can be expressed as 29 + 19 + 13

**Ex 3.2 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) 5 + 5 = 10

(iii) 2 + 13 = 15

(iv) 11 + 9 = 20

(v) 17 + 3 = 20

(vi) 7 + 13 = 20

**Ex 3.2 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