Question

Show that any positive odd integer is of the form 6q + 1, or 6q + 3, or 6q + 5, where q is
some integer

Answer 1

Answer:

Step-by-step explanation:

To Show :-

Any positive odd integer is of the form 6q + 1, or 6q + 3, or 6q + 5.

Solution :-

Let the given integer be a.

On dividing a by 6 , we get q as the quotient and r as the remainder  

a = 6q + r

r = 0, 1, 2, 3, 4, 5

a = 6q , Even Number

when r = 1

a = 6q + 1, Odd Number

when r = 2

a = 6q + 2, Even Number

when r = 3

a=6q + 3, Odd Number

when r = 4

a=6q + 4, Even Number

when r = 5,

a= 6q + 5 , Odd Number

Hence Proved.

Any positive odd integer is of the form 6q + 1, 6q + 3 or 6q + 5.

Answer 2

Step-by-step explanation:

To Show -

Any positive odd integer is of the form 6q + 1, or 6q + 3, or 6q + 5, where q is some integer

Now,

Let x be a given positive odd integer.

On dividing x by 6, let q be the quotient and r be the remainder.

Then, by Euclid's division lemma, we have

x = 6q + r, where 0 ≤ r < 6

= x = 6q + r, where r = 0,1,2,3,4,5 ...(i)

Case 1.

When r = 0

Putting r = 0 in (i), we get :

x = 6q (Even)

Case 2.

When r = 1

Putting r = 1 in (i), we get :

x = 6q + 1 (Odd)

Case 3.

When r = 2

Putting r = 2 in (i), we get :

x = 6q + 2 (Even)

Case 4.

When r = 3

Putting r = 3 in (i), we get :

x = 6q + 3 (Odd)

Case 5.

When r = 4

Putting r = 4 in (i), we get :

x = 6q + 4 (Even)

Case 6.

When r = 5

Putting r = 5 in (i), we get :

x = 6q + 5 (Odd)

Thus,

when x is odd, it is if the form (6q + 1) or (6q + 3) or (6q + 5) for some integer q.