Prove that a positive integer $n \geq 2$ is prime if and only if there is no positive integer greater than $1$ and less than or equal to $\sqrt{n}$ that divides $n$

Proof given by contradiction.

Step-by-step explanation:

Given that:

To prove:

is prime if and only if no positive integer > 1 and divides .

Solution:

First of all, let is a composite number i.e. not a prime number such that:

and and are prime and divides and also divides .

Let

or

1. :

is prime and is a divisor of .

2. :

We have assumed that

is a prime number and is a divisor of .

But we are given that no prime number divides but we have proved that divides .

So, it is a contradiction to our assumption.

Therefore, our assumption is wrong that   is a composite number.

Hence, proved that is a prime number.

it would open into a square depending on the type of envelope i’m not sure about the second part

multiplication is multiplication.   it might feel harder depending on what level of math you are in, but generally it will be the same old stuff, while implementing some new concepts and methods of solving work.