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:

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. 

Proof: Let n = 6q + 5

Since any positive integer n is of the form of 3k or 3k + 1, 3k + 2

If q = 3k

Then, $n=6 q+5$

 

$\begin{aligned} \Rightarrow \quad n &=18 k+5(q=3 k) \\ \Rightarrow \quad n &=3(6 k+1)+2 \\ \Rightarrow \quad n &=3 m+2(\text { where } m=(6 k+1)) \end{aligned}$

If q = 3k + 1

Then, $n=(6 q+5)$

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

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

 

$\Rightarrow \quad n=18 k+11$

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

 

$\Rightarrow \quad n=3 m+2($ where $m=(6 k+3))$

If q = 3k + 2

Then, $n=(6 q+5)$

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

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

$\Rightarrow \quad n=18 k+17$

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

 

$\Rightarrow \quad n=3 m+2($ where $m=(6 k+5))$

Consider here 8 which is the form 3q + 2 i.e. 3 × 2 + 2 but it can’t be written in the form 6q + 5. Hence the converse is not true