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


Answer:


Step by Step Explanation:
  1. Let n be a positive integer such that any prime number less than or equal to n does not divide n.
    Now, we have to prove that n is prime.
  2. Let us assume n is not a prime integer, then n can be written as
    n=yz where 1<yz
    yn and zn
  3. Let p be a prime factor of y, then, pyn and p divides y.
    p|yzp|n.....(1)
  4. By eq(1), we get a prime number less than or equal to n that divides n. This contradicts the given fact that any prime number less than or equal to n does not divide n, therefore, our assumption that n is not a prime integer was wrong.
  5. Hence, if no prime number less than or equal to n divides n, then n is a prime integer.

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