Question

Design an algorithm that is given a positive integer \(N\) and determines whether \(N\) is a prime number, that is, not evenly divisible by any value other than 1 and itself. The output of your algorithm is either the message 'not prime', along with a factor of \(N\), or the message 'prime'.

Short answer

Check divisibility up to \(\sqrt{N}\); return 'not prime' with a divisor if divisible, otherwise 'prime'.

Step-by-step solution

Understand the Definition of a Prime Number

A prime number is a positive integer greater than 1 that has no positive divisors other than 1 and itself. Therefore, to determine if a given number \(N\) is prime, we must check if \(N\) is divisible by any number other than 1 and \(N\) itself.

Establish the Range for Checking Divisors

To check if \(N\) is a prime number, it's sufficient to test for divisibility by any integer from 2 up to the square root of \(N\). This is because if \(N = a \times b\) and both \(a\) and \(b\) are greater than the square root of \(N\), then \(a \times b\) exceeds \(N\), which is not possible. Thus, we only need to check divisors up to \(\sqrt{N}\).

Implement the Prime Checking Algorithm

The algorithm involves the following steps:1. If \(N\) is less than or equal to 1, it is not prime.2. For each integer \(i\) from 2 up to \(\sqrt{N}\), check if \(N\) is divisible by \(i\) (i.e., \(N \mod i = 0\)).3. If there is any \(i\) that divides \(N\) evenly, output 'not prime' and the divisor \(i\).4. If no divisors are found, output 'prime'.

Coding the Prime Check

Here is a pseudocode representation of the algorithm:```function isPrime(N) if N <= 1 return 'not prime' end if for i from 2 to sqrt(N) if N mod i == 0 return 'not prime', i end if end for return 'prime'end function```This function receives a positive integer \(N\) and evaluates whether it is prime by checking divisibility through potential factors up to \(\sqrt{N}\).

Key concepts

Prime Numbers

Prime numbers are fundamental in mathematics. They are unique because they can only be divided by 1 and themselves without leaving a remainder. This means their only factors are 1 and the number itself.
Imagine you have beads and you're creating necklaces. If you can only make a necklace of one bead or one that includes all the beads, the number of beads might be a prime number. To further explain:

• A prime number is greater than 1.
• Examples include 2, 3, 5, 7, and 11.
• Non-prime numbers, known as composite numbers, have more divisors, such as 4 (with divisors 1, 2, 4) or 6 (with divisors 1, 2, 3, 6).

Understanding prime numbers helps in various fields, such as cryptography and computer algorithms, where security and efficient problem-solving are crucial.

Divisibility

Divisibility is a key concept when determining if a number is prime. It refers to one number being able to be divided by another without leaving a remainder. For example, 10 is divisible by 2, 5, and 1 because dividing it by these numbers leaves no remainder. To check if a number, say 15, is divisible by another number, say 3, you would perform the division 15 divided by 3 and see if it results in a whole number. If it does, 3 is a divisor of 15. Some important points about divisibility:

• Every number is divisible by 1 and itself.
• If a number is divisible by any number other than 1 and itself, it's not a prime number.
• In our algorithm, we test divisibility for numbers between 2 and the square root of the given number (N) to efficiently determine primality.

Recognizing patterns in divisibility helps simplify complex calculations and is essential in algorithm design.

Pseudocode

Pseudocode is a way to describe an algorithm using a set of instructions written in plain language. It doesn't require specific programming syntax but follows logical steps that a computer could execute if translated into code. Think of pseudocode as instructions you might give a friend to make a sandwich: straightforward, step-by-step, and avoids technical jargon. Here's why pseudocode is helpful:

• It focuses on the logic of solving a problem without getting bogged down by coding details.
• It can be translated into any programming language.
• It allows you to map out complex algorithms simply and understandably.

In our primality test example, the pseudocode checks:
- If the number is less than or equal to 1, it’s not prime.
- For each number from 2 to the square root of the number, check divisibility.
- If divisible evenly by any, it's not prime; otherwise, it is. Using pseudocode helps you design efficient algorithms logically and cohesively.