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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.


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=18q+5=18q+3+2
n=3(6q+1)+2
n=3m+2 where m=6q+1

Now, if q=(3k+1)
n=6(3k+1)+5
n=18q+6+5
n=18q+9+2
n=3(6q+3)+2
n=3m+2 , where m=6q+3

Now , if q=3k+2
n=6(3k+2)+5
n=18q+12+5
n=3(6q+5)+2
n=3m+2 , where m=(6q+5)

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
Now, if q=6q
n=3q+2
n=3(6q)+2
n=18q+2
n=2(9q+1)
n=2m
Here clearly we can observe that 3q+2 is not in the form of 6q+5.

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

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