Question

I need help with a problem of coprime numbers solution (I don't fully understand it).
Problem: Prove that two consecutive odd numbers are coprime between themselves.
Solution: Let's say that odd numbers are 2k+3 and 2k+1. If they had a common divisor, then their difference could be divided by such number as well. (2k+3)-(2k+1)=2. Number 2 isn't their common divisor because they are odd. That means, they have only one common divisor -1. That's why they are coprime.

Answer 1

What I don't understand: Why does this hold true? "If they had a common divisor, then their difference could be divided by such number".
Could anyone explain / point out theorem that shows that this is true?

Answer 2

I think the reasoning is something like this. Suppose that they had a common divisor, say "d". Then, we could write:
2k + 3 = d*A,
2k + 1 = d*B,
where A and B are some other numbers (other factors).

Then, the difference can be divided by this common divisor as seen here:
(2k+3)-(2k+1) = d*A - d*B = d*(A-B)
which is clearly divisible by d.

However, the solution points out that
2 = d*(A-B)

But this is a big problem! If d = 2, then 2k+3 = 2*A is even! (and likewise for 2k+1). So that is impossible. So, d must be 1, and A = 2k+3, B = 2k+1.

In other words, the only common divisor is 1, which is the definition of coprime.

Answer 3

Thank you very much! I didn't think of rewriting the odds that way... You helped me a bunch!

Answer 4

Yep, it's a little trick, but it pays off. :)