In this problem, we study the relationship between two algebraic forms of a positive integer. We are required to prove that every integer of the form 6q + 5 can also be written in the form 3q + 2 for some integer q, but the converse statement is not true.
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 the given positive integer be of the form
6q + 5, where q is an integer.
We can rewrite this expression as follows:
6q + 5 = 3(2q + 1) + 2
Let p = 2q + 1.
Since q is an integer, p is also an integer.
Therefore,
6q + 5 = 3p + 2
Hence, any positive integer of the form 6q + 5 can be written in the form 3p + 2 for some integer p.
This proves the first part of the statement.
Now, we show that the converse is not true.
Consider a number of the form 3q + 2.
Take q = 0.
Then,
3q + 2 = 2
But 2 cannot be written in the form 6q + 5 for any integer q.
Hence, a number of the form 3q + 2 need not be of the form 6q + 5.
Conclusion
Therefore, every positive integer of the form 6q + 5 is also of the form 3q + 2 for some integer q, but the converse statement is not true.
Hence proved.