Question

A positive whole number \(n>2\) is prime if no number between 2 and \(\sqrt{n}\) (inclusive) evenly divides \(n .\) Write a program that accepts a value of \(n\) as input and determines if the value is prime. If \(n\) is not prime, your program should quit as soon as it finds a value that evenly divides \(n\).

Short answer

Check divisibility up to \(\sqrt{n}\); if none divides evenly, \(n\) is prime.

Step-by-step solution

Understand the Problem Statement

We need to create a program that checks if a given number \(n\) greater than 2 is prime. The program should determine if any number between 2 and \(\sqrt{n}\) divides \(n\) without a remainder. If such a number is found, \(n\) is not prime.

Input and Initial Check

Accept the input value of \(n\). Ensure \(n > 2\) since a prime number must be greater than 2.

Edge Case Handling

If the input \(n\) is an even number greater than 2, it's not prime as it is divisible by 2. Print that \(n\) is not prime and exit.

Iterate from 3 to \(\sqrt{n}\)

Initialize a loop starting from 2 up to the integer value of \(\sqrt{n}\). For each number in this range, check if it divides \(n\) evenly.

Division Check

For each integer divisor in the loop, perform \(n \mod \text{divisor}\). If the remainder is zero, print that \(n\) is not a prime number and exit the loop.

Declaration of Primality

If no divisors are found that evenly divide \(n\) within the loop, declare \(n\) as a prime number.

Key concepts

Python programming

In this task, Python programming helps us to automate the process of checking whether a number is prime. Python, with its simple syntax and vast range of libraries, is a perfect choice for mathematical calculations. To solve this problem, we need to follow these basic steps:

• Collect the user's input using the input function.
• Convert the input to an integer since mathematical operations require numerical data types.
• Perform arithmetic and logical operations to check for primality using loops and conditions.
• Output the result based on the conditions you've programmed. If a number divides evenly, it's not prime.

Python's rich set of control structures, like loops and if conditions, make it comfortable to express such algorithms clearly and concisely. Moreover, Python provides functions like `math.sqrt()` from the math library to compute square roots, which can be particularly handy in such problems to determine the range of potential divisors.

number theory

Number theory is an essential mathematical concept that helps us understand how numbers behave, especially divisibility, which is at the core of checking primes. A prime number is one of the simplest concepts but fundamental in number theory. It's defined as a number that only has two distinct positive divisors: 1 and itself.

For example, the number 7 is prime because only 1 and 7 divide it without a remainder. In the context of algorithms, understanding that we only need to check divisors up to the square root of a number ( ext{n} ) drastically reduces the amount of work needed.

The insight here is that if a number `n` is divisible by some number greater than its square root, there must be a corresponding factor smaller than the square root. This significantly optimizes our algorithm making it efficient to implement in a programming language like Python.

input validation

Input validation is crucial when writing a program that requires user input. In this case, we need to ensure that the value entered by the user for `n` is a positive integer greater than 2. Input validation prevents unwanted errors from improper data.

Here's how you can do it in Python:

• Firstly, use the `input()` function to get the raw input from the user.
• Then, convert this input to an integer using `int()`. This step automatically checks if the input is numeric.
• Next, use an `if` statement to check if the number is greater than 2. If not, you may want to print an error message and stop the program or ask for the input again.

Ensuring that the input is correct before proceeding with calculations avoids runtime errors and ensures that your algorithm gets the proper data to work with, leading to accurate results.

conditional loops

Conditional loops are an integral part of algorithms that involve repetitive checks until a specific condition is met. In our prime-checking algorithm, conditional loops help determine whether a number `n` has any divisors between 2 and its square root.

**For Loop in Python:**
You can implement a conditional loop using the `for` loop, iterating over a range from 2 to the integer square root of `n`, checking divisibility.

• Start the loop from a divisor of 2.
• Use `range(2, int(math.sqrt(n))+1)` to limit the loop, as factors do not exist beyond the square root for composite numbers.
• Inside the loop, use an `if` statement to check divisibility using the modulus operator `%`.

If you find a divisor where `n % divisor == 0`, it signals that `n` is not prime, and the loop can be exited, usually with a `break` statement. This design offers efficiency as it ceases to check further divisors once a non-primality condition is detected.