The smallest value of natural number $n$, for which $2n + 1$ is not a prime number, is

### Concept & Strategy Evaluate the given algebraic expression by substituting consecutive natural numbers starting from the smallest ($n = 1$) until the result evaluates to a composite (non-prime) number. ### Step-by-Step Solution * **Given Expression:** $$E = 2n + 1$$ where $n$ is a natural number ($n = 1, 2, 3, \dots$). * **Test Values:** Let's substitute natural numbers sequentially: For $n = 1$: $E = 2(1) + 1 = 3$ (Prime) For $n = 2$: $E = 2(2) + 1 = 5$ (Prime) For $n = 3$: $E = 2(3) + 1 = 7$ (Prime) For $n = 4$: $E = 2(4) + 1 = 9$ (Composite, divisible by 3) * The condition is met at $n = 4$, which results in 9 (not a prime number). ### Exam Strategy & Shortcut Rather than testing all numbers from $n = 1$, you can directly use the provided options. Test $n = 3$ first, which yields 7 (prime). Next test $n = 4$, which yields 9 (not prime). You have found the answer instantly. ### Common Pitfall Confusing "natural numbers" (which start from 1) with "whole numbers" (which start from 0). If you plug in $n = 0$, you get 1, which is not prime, but 0 is not a natural number. Another mistake is forgetting the prompt asks for when the result is *not* prime. ### Final Answer **Therefore, the correct answer is 4.**