Prove that if a positive integer is of the form $6q+5$, then it is of the form $3q+2$ for some integer q, but not conversely.

Given: Positive integer of the form $6q\ +\ 5$.

To prove: Here we have to prove that if a positive integer is of the form $6q\ +\ 5$, then it is of the form $3q\ +\ 2$ for some integer q, but not conversely.

Solution:

Let, $n\ =\ 6q\ +\ 5$, where q is a positive integer.

We know that any positive integer is of the form $3k$, $3k\ +\ 1$, $3k\ +\ 2$.

Now,

If $q\ =\ 3k$ then,

$n\ =\ 6(3k)\ +\ 5$

$n\ =\ 18k\ +\ 5$

$n\ =\ 18k\ +\ 3\ +\ 2$

$n\ =\ 3(6k\ +\ 1)\ +\ 2$

$n\ =\ 3m\ +\ 2$

Where $m\ =\ 6k\ +\ 1$ and is an integer.

If $q\ =\ (3k\ +\ 1)$

$n\ =\ 6(3k\ +\ 1)\ +\ 5$

$n\ =\ 18k\ +\ 6\ +\ 5$

$n\ =\ 18k\ +\ 9\ +\ 2$

$n\ =\ 3(6k\ +\ 3)\ +\ 2$

$n\ =\ 3m\ +\ 2$

Where $m\ =\ 6k\ +\ 3$ and is an integer.

If $q\ =\ 3k\ +\ 2$

$n\ =\ 6(3k\ +\ 2)\ +\ 5$

$n\ =\ 18k\ +\ 12\ +\ 5$

$n\ =\ 3(6k\ +\ 5)\ +\ 2$

$n\ =\ 3m\ +\ 2$

Where $m\ =\ 6k\ +\ 5$ and is an integer.

Therefore, if a positive integer is of the form $6q\ +\ 5$ then it is of the form $3q\ +\ 2$.

Now, let $n\ =\ 3q\ +\ 2$, where q is a positive integer.

We know that any positive integer is of the form $6q$, $6q\ +\ 2$, $6q\ +\ 3$, $6q\ +\ 4$, $6q\ +\ 5$

If $q\ =\ 6k$,

$n\ =\ 3q\ +\ 2$

$n\ =\ 3(6k)\ +\ 2$

$n\ =\ 18k\ +\ 2$

$n\ =\ 2(9k\ +\ 1)$

$n\ =\ 2m$

Here we can clearly see that $3q\ +\ 2$ is not in the form of $6q\ +\ 5$.

Hence, it can be concluded that if a positive integer is of the form $6q\ +\ 5$, then it is of the form $3q\ +\ 2$ but not conversely.