1. **Problem Statement:** We want to determine for which values of $n$ the product $$\left(1 + \frac{1}{2}\right)\left(1 + \frac{1}{3}\right)\left(1 + \frac{1}{4}\right) \ldots \left(1 + \frac{1}{n}\right)$$ is an integer. 2. **Rewrite each term:** Each term inside the product can be rewritten as $$1 + \frac{1}{k} = \frac{k+1}{k}$$ for $k = 2, 3, \ldots, n$. 3. **Express the product:** The entire product becomes $$\prod_{k=2}^n \frac{k+1}{k} = \frac{3}{2} \times \frac{4}{3} \times \frac{5}{4} \times \cdots \times \frac{n+1}{n}$$ 4. **Simplify the telescoping product:** Notice that all intermediate terms cancel out: $$= \frac{3}{\cancel{2}} \times \frac{\cancel{4}}{3} \times \frac{5}{\cancel{4}} \times \cdots \times \frac{n+1}{n} = \frac{n+1}{2}$$ 5. **Determine when the product is an integer:** The product equals $$\frac{n+1}{2}$$ which is an integer if and only if $n+1$ is divisible by 2, i.e., $n+1$ is even. 6. **Analyze parity:** Since $n+1$ is even, $n$ must be odd. **Final answer:** The product is an integer when $n$ is odd. **Answer choice:** A) odd