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 A = 6q + 5, be any number, where q is any positive integer.

Part 1:

To show A is in the form of 3q + 2, where q is another integer A = 6q + 5 = 6q + 3 + 2 = 3(2q + 1) + 2 = 3q' + 2

As,

q is any positive integer, q' = 3q + 2 is also a positive integer and hence 6q + 5, is in form of 3q' + 5

Part 2:

To show converse is not true, i.e. if a no is in the form of 3q + 2, then it may or may not be in the form of 6q + 5

For example, consider:

8 = 3(2) + 2 is in the 3q + 2 form, but it can't be expand in 6q + 5 form.