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

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Answer

Answer

Proof:
To prove that a positive integer n is a prime number, we need to show that no prime p less than or equal to √n divides n.

Assumption:
Let's assume that n is not a prime number. In other words, n is a composite number, which means it can be expressed as the product of two positive integers a and b, where 1 < a='' ≤='' b='' />< />

Divisors of n:
Since n is a composite number, it has divisors other than 1 and n. Let's consider a divisor d of n such that 1 < d='' ≤='' />

Case 1: d ≤ √n
If d is less than or equal to √n, then a and b both must be greater than √n because a ≤ b. It implies that a * b > (√n) * (√n) = n, which contradicts our assumption that a * b = n. Therefore, there can be no divisor d ≤ √n.

Case 2: d > √n
If d is greater than √n, then a and b both must be less than √n. However, in this scenario, a * b < (√n)='' *='' (√n)='n,' which='' again='' contradicts='' our='' assumption='' that='' a='' *='' b='n.' thus,='' there='' can='' be='' no='' divisor='' d='' /> √n.

Conclusion:
Since there are no divisors of n that are less than or equal to √n, we can conclude that n is not a composite number. Hence, n must be a prime number.

Example:
Let's take an example to illustrate the proof. Consider n = 17.

- The square root of 17 is approximately 4.123.
- The prime numbers less than or equal to 4 are 2 and 3.
- If we divide 17 by 2 or 3, we get remainders of 1 and 2, respectively. Therefore, no prime less than or equal to √17 divides 17.
- Hence, 17 is a prime number, which validates the proof.

Summary:
In summary, we have proven that a positive integer n is a prime number if no prime p less than or equal to √n divides n. This proof is based on the assumption that if n is a composite number, it can be expressed as the product of two positive integers. By considering all possible cases, we have shown that there can be no prime divisor of n that is less than or equal to √n. Therefore, n must be a prime number.

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