# The only positive divisor of both $a$ and $a + 1$ is $1$

Prove that if $a \in \mathbb Z$ then the only positive divisor of both $a$ and $a + 1$ is $1$.

When I saw this statement I didn't understand it. The only way that I can see it being true is if a is a negative number but since a ∈ Z a could also be positive and thus it could have more than one positive divisor. If that's the case would I have to disprove this instead of proving? And what would be the best way to do that?

• What common divisors do 2 and 3 have? Or 3 and 4 or 9 and 10? – Paul Sundheim Sep 22 '14 at 0:18
• I think it's asking for a single number that divides both $a$ and $a+1$. – Trurl Sep 22 '14 at 0:19
• Please choose descriptive titles. The original title, Prove or Disprove??? carried virtually no information of what the problem is about. – user147263 Sep 22 '14 at 1:11

You seem to have misunderstood the question. As far as I can tell you are thinking about divisors of $a$ and $a+1$ separately, but the question is asking you to show that the only positive number which is a divisor of both $a$ and $a+1$ simultaneously is $1$.
If $n$ divides both $a$ and $a+1$, it would also divide $a+1-a$.