Question

Prove that a positive integer n is prime number,if no prime p less than or equal to √n divides n

Answer 1

Let any integer n ≥ 0 is a composite number.
so, n has a factor between 1 to n.

Let r is the factor of n, such that, 1 < r < n

so, we can write n = rs , where r and s are positive integers such that, 1 < r, s < n

assume , integer r is greater than equal to s.
e.g., r ≤ s

And also consider s > √n

so, √n < s ≤ r

it means, r > √n

but n = rs > √n. √n

so, n > n which is a contradiction.
hence, our assumption was wrong.
therefore, a positive integer n is prime number,if no prime p less than or equal to √n divides n

Answer 2

Hope it help you

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