Question

Write a program that takes a whole number input from a user and determines whether it is prime. If the number is not prime, display its unique prime factors. Remember that a prime number's factors are only 1 and the prime number itself. Every number that is not prime has a unique prime factorization. For example, consider the number \(54 .\) The prime factors of 54 are 2,3,3 and 3\. When the values are multiplied together, the result is \(54 .\) For the number \(54,\) the prime factors output should be 2 and \(3 .\) Use Sets as part of your solution.

Short answer

Check if the input is prime; otherwise, display unique prime factors using a set.

Step-by-step solution

Input the Number

Prompt the user to input a whole number. This number will be used to determine whether it is a prime number or not.

Check if the Number is Prime

Develop a function to check whether the given number is prime. A prime number is only divisible by 1 and itself. The function should return true if the number is prime and false otherwise. For instance, check divisibility from 2 up to the square root of the number.

Find Prime Factors

If the number is not prime, develop a function to find its prime factors. Start with the smallest prime, 2, and divide the number by smallest prime factors, adding each to a list if completely divisible, until the remaining part of the number is 1.

Convert List to Set for Unique Factors

Convert the list of prime factors into a set to ensure only unique factors are displayed. For example, if the number is 54, the prime factors are [2, 3, 3]. By converting to a set, it becomes {2, 3}.

Display Result

If the number is prime, display a message stating it is a prime number. If the number is not prime, display its unique prime factors, which are stored in the set.

Key concepts

Prime Number Identification

To determine if a number is prime, we must first understand what makes a number prime. A prime number is a natural number greater than 1 that has no positive divisors other than 1 and itself. This means it cannot be divided evenly by any other numbers. We often start checking for divisibility from 2 up to the square root of the number, because if a number is divisible by any integer greater than its square root, it would have to be divisible by an integer less than its square root. For example, to check if 29 is prime, we would only need to test divisibility with numbers up to 5 (since the square root of 29 is approximately 5.39). If it is not divisible by any of these, it's prime.
In programming, the common approach is to create a function that loops through potential divisors to check this condition. If none divide the number without a remainder, it’s confirmed as a prime.

Prime Factorization

When a number is not prime, it has a prime factorization—a way of expressing it as a product of prime numbers. For instance, the number 54 can be expressed as the product of prime numbers: 2 and 3. Specifically, the prime factorization of 54 is 2 × 3 × 3 × 3. Yet, we only list each prime once to show unique factors using sets.
To find the prime factors, start with the smallest prime number, 2, and divide the target number by 2 repeatedly until no longer possible. Then, move to the next smallest prime number and repeat the process until the target reduces to 1.
Thus, any composite number can be decomposed into its prime factors through repeated division and can be verified easily by reconstructing the product to see if it matches the original number.

Set Data Structure

In programming, using a set data structure helps us manage a collection of unique elements. It's particularly useful in scenarios where we need to store a collection of prime factors as seen in prime factorization. A set automatically handles duplicate entries and ensures that each element is unique.
For example, if the prime factors of a number are [2, 3, 3], converting this list into a set would result in {2, 3}. This transformation removes the duplicate 3, ultimately helping to present a simpler format by showing only the fundamental prime factors.

• Sets are typically unordered and accessed faster than lists, making them suitable for searching and retrieving elements quickly.

This feature of sets simplifies both the coding and the output process when working with mathematical problems involving factors.

Mathematical Programming Concepts

Creating algorithms for mathematical checks is a key part of programming. We apply logic to repeat patterns that can solve mathematical operations consistently. For example, identifying whether a number is prime or finding its factors are built upon the same foundational programming concepts.
One constructs functions for specific purposes, like detecting primality or breaking down a number into prime factors. These functions often incorporate loops, conditional statements, and data structures such as lists and sets.

• Loops help iterate through potential divisors up to the square root of a number.
• Conditionals determine whether division results in zero remainder.
• Data structures manage and store results efficiently.

Programming not only automates such calculations but also improves accuracy and speed, allowing for practical applications in larger datasets or more complex mathematical problems.