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The relation R on the set {1, 2, 3, ..., 10} defined by R = {(x, y) : x and y have exactly one common factor other than 1} is

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1. **Test Reflexive**: For $(x, x)$ to be in $R$, $x$ must have exactly one factor other than 1. This would only be true for prime numbers. For $x=4$, factors are $\{1, 2, 4\}$ (two factors other than 1). Thus, not reflexive for all $x$.
2. **Test Symmetric**: If $(x, y)$ is a pair where they share exactly one prime factor, then surely $(y, x)$ shares that same property. **Symmetric** (Yes).
3. **Test Transitive**: 
   - Let $x=4, y=6$. Common factor is $\{2\}$. So $(4,6) \in R$.
   - Let $y=6, z=9$. Common factor is $\{3\}$. So $(6,9) \in R$.
   - For transitivity, $(4,9)$ must be in $R$. But $4, 9$ have **zero** common factors other than 1. Not transitive.
4. **Conclusion**: Symmetric only.

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