**Telangana**** SCERT Solution Class IX (9) Math Chapter 15 Proofs in Mathematics Exercise 15.3**

**1.)**

(i) From the given examples we can make the following conjecture:

(a)The product of any three consecutive odd number is odd.

(b) The product of any three consecutive odd number is divisible by 3 since, all three sums 15, 105, 210, is divisible by 3.

(c) Sum of all the digits present in product of three consecutive odd number is even.

(ii) From the given example we can conclude that

(a) The sum of any of three consecutive even numbers is even.

(b) The sum of any of three consecutive even numbers is always divided by 3 in this case 12, 24, 30, are all divided by 3.

(c) The sum of any three consecutive even numbers is divisible by 6.

**3.)** From the given example we see that the numbers are divided by multiplying its prime factors.

Now,

From the two examples the pattern we can see is that the total number of factors is the multiplication of the sum of each factors and the no times it repeats itself.

For eg.

28 = 2^{2} x 7^{1}

Here 2, 7 are prime factors of 28.

2 has power 2 & 7 has 1

So, (1+2) x (1+4) = 6 as the total no of factors.

**4.**) From the pattern we can see that 1^{2} = 1

(a) Each time the no of digits is increased by 11^{2}= 121

111^{2} = 12321

(b)The number is a palindrome number 1111^{2} = 12344321

Which goes as much as the number of 11111^{2} = 123454321

Digits in the number to be squared as 11^{2} goes upto 2 since 11 has two digit 1 ,1.

(c) The number repeats itself reversing backwards where it reaches the highest number.

111111^{2} = 12345654321

1111111^{2} = 1234567654321

Yes, the conjecture is true for both.

**5.)** Axioms used in this book are

(i) Lines are infinite.

(ii) Circle can be drawn from it’s centre given its radius.

(iii) Sum of two even numbers will always be even.

(iv) Sum of two odd numbers will always be odd.

(v) Two parallel lines will make a sum of 180 ̊of its interior angles when drawn through the same line Segment.