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k0ka [10]
3 years ago
11

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 $n$
Mathematics
1 answer:
Over [174]3 years ago
7 0

Answer:

Proof given by contradiction.

Step-by-step explanation:

Given that:

n \geq 2

To prove:

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

Solution:

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

n =a\times b

and a and b are prime and a divides n and b also divides n.

Let \sqrt n = p

or n  = p\times p

1. \underline{a < p}:

a is prime and is a divisor of n.

2. \underline{a>p}:

n = a\times b = p\times p

We have assumed that a > p  \Rightarrow b

b is a prime number and is a divisor of n.

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

So, it is a contradiction to our assumption.

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

Hence, proved that n is a prime number.

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