Question

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.

Answer 1

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

we know that any positive integer is of the form 3k or, 3k + 1 or, 3k + 2

Case : 1

If q = 3k,then

n = 6q + 5

n = 6(3k) +5

n = 18k + 5  

n = 18k + 3 +2

n = 3 (6k + 1) + 2

n = 3m + 2  

[where m = (6k + 1)]

Case : 2

If q = 3k+ 1,then

n = (6q + 5)

n = (6 (3k + 1) + 5)    

n = 18k + 6 + 5

n = 18k + 11

n = 18k + 9 +2

n = 3 (6k + 3) + 2

n = 3m + 2  

[where m = (6k + 3)]

Case : 3

If q = 3k + 2,then

n = (6q + 5)

n = (6 (3k + 2) + 5)  

n = 18k + 12 + 5

n = 18k + 17

n = 18k + 15 + 2

n = 3 (6k + 5) + 2

n = 3m + 2  

[where m = (6k + 5)]

Hence, if a positive integer is of the form (6q + 5),then it is of the form 3q+2 for some integers q.

Conversely :

Let n = 3q +2

We know that a positive integer can be of the form 6k + 1, 6k + 2, 6k + 3, 6k + 4 or 6k + 5

Case : 1

if q = 6k + 1, then

n = 3(6k + 1) + 2

n = 18k + 3 + 2

n= 18k + 5

= 6(3k) + 5

= 6m + 5, where m= 3k  

Case : 2

if q = 6k + 2 then,

n = 3(6k + 2) + 2

n = 18k + 6+ 2

n = 6 (3k + 1) + 2

n = 6m + 2, where m = (3k+1)

Now, this is not of the form 6m + 5

Hence, if n is of the form 3q + 2, then it is not necessary that it be of the form 6q +5 .

Hence the converse is not true.

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