Question

Prove that if a positive integer is of the form 6q+5then it is of form 3q+2 for same integer q

Answer 1

Let us consider any number of the form N = 6q + 5 = 3*2q +3 + 2 = 3*(2q+1) + 2 = 3q' +2 where q' is equal to 2q +1 which will be a positive interger since q is also an integer. whereas if the number is of the form N = 3q+2 = 3*(q'+1) + 2 = 3q'+5 = 6*q'/2 +5 here since q is an integer then q-1 = q' is also an integer but it does not guarantee that q'/2 is also an integer therfore the converse is not definitely true.