Prove that a positive integer ^@n^@ is a prime number if no prim | Math Question and Answer

Answer:

Step by Step Explanation:

Let $n$ be a positive integer such that any prime number less than or equal to $\sqrt{n}$ does not divide $n$ .
Now, we have to prove that $n$ is prime.
Let us assume $n$ is not a prime integer, then $n$ can be written as
$n = yz$ where $1 < y \le z$
$\implies y \le \sqrt{n}$ and $z \ge \sqrt{n}$
Let $p$ be a prime factor of $y$ , then, $p \le y \le \sqrt{n}$ and $p$ divides $y$ .
$\begin{align} \implies & p | yz \\ \implies & p | n && .....(1) \end{align}$
By eq(1), we get a prime number less than or equal to $\sqrt{ n }$ that divides $n$ . This contradicts the given fact that any prime number less than or equal to $\sqrt{n}$ does not divide $n$ , therefore, our assumption that $n$ is not a prime integer was wrong.
Hence, if no prime number less than or equal to $\sqrt{n}$ divides $n$ , then $n$ is a prime integer.

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