Exercise 15.3

1. (i) Take any three consecutive odd numbers and find their product; for example, 1 × 3 × 5 = 15, 3 × 5 × 7 = 105, 5 × 7 × 9 = 315, 7 × 9 × 11 = 693

(ii) Take any three consecutive even numbers and add them, say, 2 + 4 + 6 = 12, 4 + 6 + 8 = 18, 6 + 8 + 10 = 24, 8 + 10 + 12 = 30 and so on.

Is there any pattern can you guess in these sums? What can you conjecture about them?

Sol:

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

(a) The product of any three consecutive odd numbers is

(b) The product of any three consecutive odd numbers is divisible by [∵ 15, 105, 315, 693 are all divisible by 3]

(c) The sum of all the digits present in product of three consecutive odd numbers is

(ii) From the given examples we can conclude that:

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

(b) The sum of any three consecutive even numbers is always divisible by [∵ 12, 18, 24, 30 are all divisible by 3]

(c) The sum of any three consecutive even numbers is always divisible by [∵ All sums are divisible by both 2 and 3]

2. Go back to Pascal's triangle.

Line-1 : 1 = 110

Line-2 : 11 = 111

Line-3 : 121 = 112

Make a conjecture about Line-4 and Line-5.

Does your conjecture hold? Does your conjecture hold for Line-6 too?

Sol:

From the given pattern, we can observe that:

Line-1: 1 = 11

Line-2: 11 = 11

Line-3: 121 = 11

∴ Conjecture for Line-4: = 11

∴ Conjecture for Line-5: = 11

Verification: {.reveal(when="blank-5")}Line-4: 1 3 3 1 = 1331 11

Line-5: 1 4 6 4 1 = 14641 11

For Line-6: 1 5 10 10 5 1 = 15101051 11

∴ The conjecture holds for Line-4 and Line-5 but does not hold for Line- [∵ The pattern breaks when we have two-digit numbers in the expansion]

3. Look at the following pattern:

i) 28 = 22 × 71

Total number of factors (2+1) × (1+1) = 3 × 2 = 6

28 is divisible by 6 factors i.e. 1, 2, 4, 7, 14, 28

ii) 30 = 21 × 31 × 51

Total number of factors (1+1) × (1+1) × (1+1) = 2 × 2 × 2 = 8

30 is divisible by 8 factors i.e. 1, 2, 3, 5, 6, 10, 15, 30

Find the pattern.

(Hint : Product of every prime base exponent +1)

Sol:

From the given examples we can observe that:

1. Numbers are expressed as products of their

2. For 28 = 22 × 71:

- Prime factors are and

- Exponents are and

- Total factors = (2+1) × (1+1) =

3. For 30 = 21 × 31 × 51:

- Prime factors are , , and

- Exponents are , , and

- Total factors = (1+1) × (1+1) × (1+1) =

∴ The pattern is: To find total number of factors, add to each exponent and multiply them together

4. Look at the following pattern:

1² = 1

11² = 121

111² = 12321

1111² = 1234321

11111² = 123454321

Make a conjecture about each of the following:

111111² = ?

1111111² = ?

Check if your conjecture is true.

Sol:

From the pattern we can observe that:

(a) Each time the number of digits is increased by

(b) The number is a number which goes as much as the number of digits in the number to be squared

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

∴ Conjecture for 111111² =

∴ Conjecture for 1111111² =

∴ The conjecture is for both cases [∵ The pattern follows the same rule of increasing digits and palindromic structure]

5. List five axioms (postulates) used in this book.

Sol:

The five axioms used in this book are:

(i)

(ii)

(iii)

(iv)

(v)

6. In a polynomial p(x) = x² + x + 41 put different values of x and find p(x). Can you conclude after putting different values of x that p(x) is prime for all? Is x an element of N? Put x = 41 in p(x). Now what do you find?

Sol:

Given polynomial: p(x) = x² + x + 41

Let's evaluate p(x) for different values of x:

When x = 0: {.reveal(when="blank-0")}p(0) = 0² + 0 + 41 = (prime)

When x = 1: {.reveal(when="blank-1")}p(1) = 1² + 1 + 41 = (prime)

When x = 2: {.reveal(when="blank-2")}p(2) = 2² + 2 + 41 = (prime)

When x = 3: {.reveal(when="blank-3")}p(3) = 3² + 3 + 41 = (prime)

When x = 4: {.reveal(when="blank-4")}p(4) = 4² + 4 + 41 = (prime)

When x = 41: {.reveal(when="blank-5")}p(41) = 41² + 41 + 41 = 41(41 + 1 + 1) = 41 × = (not prime)

∴ We can conclude that: {.reveal(when="blank-7")}1. p(x) is for all x ∈ N [∵ p(41) is not prime]

2. Yes, x is an element of (natural numbers)

3. When x = 41, p(x) is [∵ 1763 = 41 × 43]