Question

show that any positive odd integer is of the form 6q+1or6q+3or6q+5 where q is some integer​

Answer 1

let,

a be a positive integer,

According to Euclid division lemma,

a=bq+r

when b=6

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

Now,

integers are:-6q,6q+1,6q+2,6q+3,6q+4,6q+5

case (i)

when a=6q

=2*3q (it is an even integer with 2 as a factor)

case (ii)

when a=6q+1 (it is an odd integer as it does not have 2 as a factor)

case (iii)

when a=6q+2

=2(3q+1) ........even

case (iv)

when a=6q+3..................odd

case (v)

when a=6q+4

=2(3q+2).............even

case (vi)

when a=6q+5...............odd

thus,

with the above expanation we conclude with the fact that any positive odd integer is in the form of 6q+1,6q+3,6q+5

thank you,

hope this helps