Prove that a positive integer $n \geq 2$ is prime

Ответ:

BreBreDoeCCx

31.01.2022

Proof given by contradiction.

Step-by-step explanation:

Given that:

$n \geq 2$

To prove:

is prime if and only if no positive integer > 1 and $\leq$ $\sqrt n$ divides .

Solution:

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

$n =a\times b$

and and are prime and divides and also divides .

Let $\sqrt n = p$

or $n = p\times p$

1. $\underline{a < p}$ :

is prime and is a divisor of .

2. $\underline{ap}$ :

$n = a\times b = p\times p$

We have assumed that $a p \Rightarrow b$

is a prime number and is a divisor of .

But we are given that no prime number $\leq \sqrt n$ divides but we have proved that $b < \sqrt n$ 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.

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