Question

Show that any positive integer is of form 6q+1or6q+3 or6q+5,where q is some integer

Answer 1

Let a be the positive integer
b=6
By applying euclids division lemma
a=bq+r

If r=0
a=6q+0
a=6q=2 (3q)

If r=1
a=6q+1
a=2 (3q)+1

6q+1 is not divisible by 2

If r=2
a=6q+2
a=2 (3q+2)

6q+2 is divisible by 2


If r=3
a=6q+3
a=2 (3q)+3

6q+3y is not divisible by 2.
If r=4
a=6q+4
a=2 (3q)+4
6q+4 is divisible by 2

If r=5
a=5
a=6q+5
a=2(3q)+5
6q+5 is not divisible by 2.