Question

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

Answer 1

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.


@Altaf

Answer 2

let a be any positive integer
a=6q+r where 0< or equal to r <6
put r=1: a=6q+1 =odd integer
r=3: a=6q+3=odd integer
r=5: a=6q+5=odd integer
therefore,any positive odd integer is of the form 6q+1,6q+3 or 6q+5,where q is some integer