Question

Show that any positive odd integer is of tge form 6p+1, 6p+3, 6p+5 where p is some integer

Answer 1

Hey there!

let a be any positive integer and b=6
ie by euclids division lemma
a=bp+r,0<r<b
and p be any integer p>0
a=6p+r
where r=0,1,2,3,4,5
because 0<r<6
if a is of the form 6p,6p+2,6p+4,then a is an even integer.
as a=6p=2(3p),
or a=6p+2=2(3p+1)
or a=6p+4=2(3q+2).
a is an even integer.
but if a=6p+1=2(3p)+1=2n+1
or a=6p+3=6p+2+1=2(3p+1)+1=2n+1
or a=6p+5=6p+4+1=2(3p+2)+1=2n+1
then a is an odd positive intger


Hope it help