Given $n = 1 + x$ and $x$ is the product of four consecutive integers. Then which of the following is true? I. $n$ is an odd integer. II. $n$ is prime. III. $n$ is a perfect square.

### Concept & Logic The key insight is that the product of any four consecutive integers plus $1$ always results in the square of an odd integer. We can represent four consecutive integers as $k$, $(k+1)$, $(k+2)$, and $(k+3)$. ### Step-by-Step Solution * **Given:** $x = k(k+1)(k+2)(k+3)$ $n = x + 1$ * **Calculation / Deduction:** Group the outermost and innermost terms together to expand the expression efficiently: $x = [k(k+3)] \times [(k+1)(k+2)]$ $x = (k^2 + 3k) \times (k^2 + 3k + 2)$ Let $a = k^2 + 3k$. The equation simplifies to: $x = a(a + 2) = a^2 + 2a$ Now, calculate $n$: $n = x + 1$ $n = a^2 + 2a + 1$ $n = (a + 1)^2$ Since $n = (a + 1)^2$, $n$ is clearly a **perfect square** (Statement III is true). To check if it is odd: $a = k^2 + 3k = k(k+3)$. For any integer $k$, either $k$ is even or $(k+3)$ is even. Thus, their product $a$ is always **even**. If $a$ is even, then $(a + 1)$ is **odd**. The square of an odd number is always an **odd integer** (Statement I is true). Since perfect squares (greater than 1) have at least three factors (1, the square root, and themselves), $n$ cannot be prime (Statement II is false). ### Exam Strategy & Shortcut **Option Elimination via Substitution:** Under time pressure, pick the easiest set of consecutive integers: $1, 2, 3, 4$. * $x = 1 \times 2 \times 3 \times 4 = 24$ * $n = 1 + 24 = 25$ * Check conditions for $25$: It is odd (I is true), it is NOT prime (II is false), and it is a perfect square ($5^2$) (III is true). * Therefore, Both I and III are correct. ### Common Pitfall Students often waste time trying to algebraically prove the statements without substituting simple numbers, or they test with $0$ ($0, 1, 2, 3 \rightarrow x = 0, n = 1$), which can create ambiguity about whether $1$ acts as a prime or standard odd perfect square in this specific phrasing. Using positive consecutive integers like $1, 2, 3, 4$ is safer. ### Final Answer **Therefore, the correct answer is Both I and III are correct.**