prove That if a positive integer is of the form 6q+5 then it is of the form 3q+2 for some integer q bt not conversely
Answers
Let n= 6q+5 be a positive integer for some integer q.
We know that any positive integer can be of the form 3k, or 3k+1, or 3k+2.
∴ q can be 3k or, 3k+1 or, 3k+2.
If q= 3k, then
⇒ n= 6q+5
⇒ n= 6(3k)+5
⇒ n= 18k+5 = (18k+3)+ 2
⇒ n= 3(6k+1)+2
⇒ n= 3m+2, where m is some integer.
If q= 3k+1, then
⇒ n= 6q+5
⇒ n= 6(3k+1)+5
⇒ n= 18k+6+5 = (18k+9)+ 2
⇒ n= 3(6k+3)+2
⇒ n= 3m+2, where m is some integer
If q= 3k+2, then
⇒ n= 6q+5
⇒ n= 6(3k+2)+5
⇒ n= 18k+12+5 = (18k+15)+ 2
⇒ n= 3(6k+5)+2
⇒ n= 3m+2, where m is some integer
Hence, if a positive integer is of form 6q + 5, then it is of the form 3q + 2 for some integer q.
Conversely,
Let n= 3q+2
And we know that a positive integer can be of the form 6k, or 6k+1, or 6k+2, or 6k+3, or 6k+4, or
6k+5.
So, now if q=6k+1 then
⇒ n= 3q+2
⇒ n= 3(6k+1)+2
⇒ n= 18k + 5
⇒ n= 6m+5, where m is some integer
So, now if q=6k+2 then
⇒ n= 3q+2
⇒ n= 3(6k+2)+2
⇒ n= 18k + 6 +2 = 18k+8
⇒ n= 6 (3k + 1) + 2
⇒ n= 6m+2, where m is some integer
Now, this is not of the form 6q + 5.
Therefore, if n is of the form 3q + 2, then is necessary won’t be of the form 6q + 5.