Question

Prove that the natural number x is prime if and only if x>1 and there is no positive integer greater than 1 and less than or equal to $\sqrt{x}$ that divides x.

Answer 1

There are several ways to prove a mathematical statement. These ways include: by contradiction, by induction, contraposition, etc

How to prove the statement by contradiction

To prove the statement, we make use of proof by contradiction

From the question:

x is a composite number such that: x > 1

This means that 1 < a < x is a factor of x

This can be represented as:

$x = a * b$

Where:

• a  and b  are  positive  integers
• 1 < a, b < x

Assume that b is less than or equal to a.

Also, let

$b > \sqrt x$​

This means that:

$\sqrt x < b \ge a$

The above becomes

$\sqrt x < a$

Rewrite as:

$a > \sqrt x$

So, we have:

$x = a * b > \sqrt x * \sqrt x = x$

The above means that:

$x > x$

The above inequality is a contradiction because, a number x cannot be greater than itself

This means that, the supposition is wrong.

Hence, the given statement has been proved by contradiction

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