prove that a positive integers is of the form 6q+5,then it is of the form 3q + 2 for some integer q,but not conversely
prove that a positive integers of the form 6q+5,then it is of the form 3q + 2 for q butn't conversly
For the first part, lets assume that A is a positive integer of the form (3q+5) for some integer q
A = (6q + 5)
= (3*2q + 3 + 2)
= (3*(2q+1) + 2)
As q is an integer, then (2q +1) is also an integer, lets call it p.
= (3p + 2)
=> every positive integer of the form (6q + 5) can also be expressed as (3p + 2) for some integers q and p.
For the converse part, a simple contradiction of the claim will suffice the proof.
Consider 2 (which is a positive integer), it can be expressed in the form (3q +2) for q=0, however it can't be expressed in the form (6q + 5) for any integer q as -
2 = 6q + 5
=> q = -(1/2) which is not an integer.
Therefore, we can conclude that the converse of the given statement doesn't hold.
Show that every positive even integer is of the form2q and every positive odd
integer is of the form 2q +1where q is some integer.