Question

hey guys just simple question please solve this and send to me question is 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

To Show:-

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

AnSwer:-

Let a be a given integer.

On dividing a by 6 ,

we get q as the quotient and r as the remainder such that

★ According to Euclid Division lemma:

→ a = bq + r

Putting value of b as 6:-

a = 6q + r,

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

when r = 0,

a = 6q, even no

when r = 1

a = 6q + 1, odd no

when r = 2

a = 6q + 2, even no

when r = 3

a = 6q + 3,odd no

when r = 4

a = 6q + 4,even no

when r = 5,

a= 6q + 5 , odd no

Therefore, we can say that,

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

_______________________________

Answer 2

Let a be the positive odd integer which when divided by 6 gives q as quotient and r as remainder.

according to Euclid's division lemma

a=bq+r

a=6q+r

where , a=0,1,2,3,4,5

then,

a=6q

or

a=6q+1

or

a=6q+2

or

a=6q+3

or

a=6q+4

or

a=6q+5

but here,

a=6q+1 & a=6q+3 & a=6q+5 are odd.