Every positive odd integer is of the form 2q + 1, where ‘q’ is somea)whole numberb)natural numberc)integerd)none of theseCorrect answer is 'C'. Can you explain this answer?

Question

Every positive odd integer is of the form 2q + 1, where &#8216;q&#8217; is somea)whole numberb)natural numberc)integerd)none of theseCorrect answer is 'C'. Can you explain this answer?

Answer

Integer....euclids division lemma is applicable only for integers

Answer

Let 'a' be the positive integers and b=2

by Euclid's division algorithm
a=bq+r
a=2q + r

(I.e.) r=0 ,1
r=0 a=2q+0=> a=2q
r=1 ,a=2q +1
if a is the form of 2 m then 'a' is even integer and positive odd integer is of the form 2m +1

Answer

A = 2n + 13
B = n + 7

1) If the value of n = 1, then

A = (2*1) + 13
= 2 + 13
= 15

B = 1 + 7
= 8

Now, A = 15 and B = 8
HCF of 15 and 8 is 1.

2) If the value of n =2

A = (2*2) + 13
= 4 + 13
= 17

B = 2 + 7
= 9

A = 17 and B = 9
HCF of 17 and 9 is 1.

3) If the value of n 3

A = (2*3) + 13
= 6 + 13
= 19

B = 3 + 7
= 10

A = 19 and B = 10
HCF of 19 and 10 is 1

4) If the value of n = 4
A = (2*4) + 13
= 8 +13
= 21

B = 4 + 7
= 11

A = 21 and B = 11

HCF of 21 and 11 is 1

5) If the value of n = 5
A = (2*5) + 13
= 10 + 13
= 23

B = 5 + 7
= 12

A = 23 and B = 12
HCF of 23 and 12 is 1

6) If the value of n = 6
A = (2*6) + 13
= 12 + 13
= 25

B = 6 + 7
= 13

A = 25 and B = 13
HCF of 25 and 13 is 1

7) If the value of n = 7
A = (2*7) + 13
14 + 13
= 27

B = 7 + 7
= 14

A = 27 and B = 14
HCF of 27 and 14 is 1

8) If the value of n = 8
A = (2*8) + 13
16 + 13
= 29

B = 8 + 7
= 15

A = 29 and B = 15
HCF of 29 and 15 is 1

9) If the value of n = 9
A = (2*9) + 13
18 + 13
= 31

B = 9 + 7
= 16

A = 31 and B = 16
HCF of 31 and 16 is 1

10) If the value of n = 10
A = (2*10) + 13
20 + 13
= 33

B = 10 + 7
= 17

A = 33 and B = 17
HCF of 33 and 17 is 1

We have seen that whatever the value n has, the HCF of A and B (separate values they hold in each situation) is always 1. It means we can take any value of n and the HCF will be 1.

Answer

C) Integer

Answer

B -> natural number

Answer

Every positive odd integer is of the form 2q + 1, where q is an integer.

Explanation:
To understand why every positive odd integer can be written in the form 2q + 1, we need to understand the concept of even and odd numbers.

1. Even numbers: Even numbers are integers that are divisible by 2 without leaving a remainder. They can be expressed as 2n, where n is an integer. For example, 2, 4, 6, 8, etc., are all even numbers.

2. Odd numbers: Odd numbers are integers that are not divisible by 2 without leaving a remainder. They can be expressed as 2n + 1, where n is an integer. For example, 1, 3, 5, 7, etc., are all odd numbers.

Now, let's consider any positive odd integer. We can represent it as 2n + 1, where n is an integer. However, in this case, we want to represent it as 2q + 1.

To do this, we can substitute q = n. Since both q and n are integers, this substitution is valid. Therefore, any positive odd integer can be written as 2q + 1, where q is an integer. This is why the correct answer is option C - integer.

Let's take an example to illustrate this:

Consider the positive odd integer 7. We can express it as 2n + 1, where n = 3. However, we want to represent it as 2q + 1. So, we substitute q = n = 3, which gives us 2q + 1 = 2(3) + 1 = 6 + 1 = 7. Therefore, 7 can be represented as 2q + 1, where q is an integer.

This property holds true for all positive odd integers, as any positive odd integer can be expressed in the form 2q + 1, where q is an integer.

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