Wed April 14, 2010 By: Prateek Agarwal

prove that a positive integers of the form 6q+5,then it is of the form 3q + 2 for q butn't conversly

Expert Reply
Thu April 22, 2010

Dear student,
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.


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