Why don’t numbers with a difference of 2 have any common factors beside 1 and possibly 2? For example, if x is odd, that means x-1 and x+1 are both even.  And from what I’ve seen, because x+1 and x-1 have a difference of 2, they have no other common factors beside 1 and because they are both even, 2.  If x is even, then x+1 and x-1 are both odd and therefore have only 1 as a common factor.

My questions are:


Is this true in all cases?
If so, why is this true?
Yes, it is true in all cases. Hint: if $m$ is a common factor of $a$ and $b$, then $m$ divides $a-b$.
If $x,y$ are numbers with $d$ as common factor, then $|x-y|$ is also a multiple of $d$. Hence either $x=y$ or $|x-y|\ge d$. Therefore, numbers with $|x-y|=2$ cannot have a common factor $d$ that is $&gt;2$. Two even such numbers certainly have at least and hence exactly $2$ as common factor. Two odd numbers cannot have an even common factor, hence $1$.