Let $R$ be the relation on the set of positive integers such that $aRb$ if and only if $a$ and $b$ are distinct and have a common divisor other than 1. Which one of the following statements about $R$ is true?

Given the definition $aRb \iff a e b \text{ and } \gcd(a, b) > 1$ for positive integers $a, b$:

1. Reflexivity: For any positive integer $a$, $a$ and $a$ are not distinct ($a = a$), so $(a, a) \notin R$. Hence, $R$ is not reflexive (it is irreflexive).

2. Symmetry: If $aRb$, then $a

e b$and$\gcd(a, b) > 1$. Since$b e a$and$\gcd(b, a) = \gcd(a, b) > 1$, it follows that$bRa$. Hence,$R$ is symmetric.

3. Transitivity: Consider $a = 2$, $b = 6$, $c = 3$:

   • $a

e b$and$\gcd(2, 6) = 2 > 1 \implies 2R6$. -$b e c$and$\gcd(6, 3) = 3 > 1 \implies 6R3$. - However,$\gcd(2, 3) = 1$, so$(2, 3) \notin R$. Therefore,$R$ is not transitive.

Thus, $R$ is symmetric but not reflexive and not transitive (Option D).