Functions and Factors
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Understanding Factors
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Let's begin by discussing factors. What do we mean when we say factors of a number?
I think factors are the numbers we can multiply together to get that number.
Exactly! For example, if we take the number 18, its factors are the numbers that you can multiply to get 18, such as 1, 2, 3, 6, 9, and 18. How might we compute these in Python?
We could use a loop to check each number from 1 to n and see if it's a factor.
Correct! We can create a function that stores the factors in a list. Remember that we loop through all numbers from 1 to n, appending to our list when we find a divisor.
What's the Python code for that?
"Here's a simple function:
Identifying Prime Numbers
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What do we know about prime numbers?
Prime numbers can only be divided evenly by 1 and themselves.
Exactly! So let's leverage the `factors` function. If our list of factors for a number n is [1, n], then n is prime. How can we check this in Python?
We can call the `factors` function and see if the output equals [1, n].
"Right! Here’s a function that checks if a number is prime:
Generating Prime Numbers
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Now, how would we find all prime numbers up to n?
We could loop from 1 to n and use our `isprime` function.
"Exactly! Here's how the function can look:
Introduction & Overview
Read summaries of the section's main ideas at different levels of detail.
Quick Overview
Standard
The section explores how to compute factors of a number and checks for primality using functions in Python. It emphasizes on structuring code for reusability and maintenance, as well as the difference between using 'for' and 'while' loops, particularly in generating prime numbers.
Detailed
Functions and Factors
In programming, functions allow us to encapsulate and reuse logic efficiently. This section focuses on how to create functions in Python that compute factors and determine if a number is prime.
Key Concepts:
- Factors of a Number: The factors of a number n are integers that divide n without leaving a remainder. A simple function can compute factors from 1 to n.
- Prime Numbers: A prime number is defined as having exactly two distinct positive divisors: 1 and itself. The section provides examples of prime numbers (like 17) versus composite numbers (like 18).
- Function Composition: Functions can call other functions, illustrated through examples of
factorsandisprime, demonstrating how to break down problems into manageable code blocks. - Loops: The section highlights the use of 'for' when the range of numbers is known (like finding primes up to n) and 'while' when the number of iterations isn't predetermined (like finding the first n primes).
- Code Maintainability: Emphasizes writing understandable and maintainable code through clear structure, aiding future adjustments and optimizations.
These principles not only foster better coding practices but also enhance the efficiency and readability of the code.
Youtube Videos
Audio Book
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Understanding Factors
Chapter 1 of 6
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Chapter Content
We have already seen a function which computes the factors of a number n. So, we observed that all the factors must lie between 1 and n. This is something we can naturally compute using a for loop. We define factors of n as follows; we assume that the list of factors is empty, and for each number in the range 1 to n if that number is a divisor, is a factor of n we append it to the list of factors and eventually we return this list. This is a simple function which just takes the list of factors gives back the list of factors of the input n.
Detailed Explanation
In this chunk, we discuss how to define a function to compute the factors of a given number 'n'. Factors are the integers that divide 'n' without leaving a remainder. The process starts with an empty list where we will store these factors. We then use a for loop that iterates from 1 to 'n'. For each number in this range, we check if it divides 'n'. If it does, we add it to our list. Finally, after checking all numbers, we return this list that contains all the factors of 'n'.
Examples & Analogies
Think of it as counting how many different groups of items can be formed from a total lot. If you have 12 apples, you can take groups of 1 apple (1 factor), 2 apples (2 factors), 3 apples (3 factors), 4 apples (4 factors), and so on, up to the total (12 apples). Each group size that fits into 12 without leftovers is a factor.
Identifying Prime Numbers
Chapter 2 of 6
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Chapter Content
A prime number is one which is divisible by no other number other rather than 1 and itself. In other words, the only factors of n should be 1 and n, if n is a prime. So, 17 for example, which is a prime number has only two factors 1 and 17. Whereas, 18 which is not a prime has many more factors, it is also 2 times 9 and 3 times 6. So the list of factors of 18 is a longer list and just 1, 18. This allows us to write a very simple definition of prime based on factors which we have already seen.
Detailed Explanation
In this chunk, we introduce the concept of prime numbers. A prime number has exactly two factors: 1 and itself. We use the example of 17, which only divides evenly by 1 and 17, whereas 18 can be divided by multiple numbers such as 1, 2, 3, 6, 9, and 18. Thus, 18 is not prime. This helps us define prime numbers in terms of their factors, making it easy to check if a number is prime by filtering its factors.
Examples & Analogies
Imagine prime numbers like a solo artist: they stand alone on stage with only two fans (1 and itself), cheering just for them. In contrast, a non-prime number is like a rock band with many fans, each representing a different way to come together (their multiple factors).
Checking for Primes
Chapter 3 of 6
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Chapter Content
So what we do is we invoke the function factors and check what it returns and see if it is equal to the list 1, n. This is another illustration of the fact that if we break up our code into functions then we can use functions one inside the other, and break up our problem into smaller units which are easier to digest and to understand.
Detailed Explanation
To check if a number is prime, we call our earlier defined function that computes factors and check whether the result equals the list containing just 1 and the number itself. This demonstrates the importance of modular programming, where breaking down the problem into smaller functions makes the code easier to navigate and understand.
Examples & Analogies
Think about cooking a complicated recipe. Instead of preparing everything at once, you can separate the tasks: chopping, boiling, and frying. After each simple task, you get a clearer view of what you need for the entire dish. Similarly, using smaller functions in programming lets us handle complex problems more systematically.
Boundary Conditions with Prime Numbers
Chapter 4 of 6
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Chapter Content
One small thing to be aware of when we are dealing with prime numbers is that we should not accidentally define 1 to be a prime, because if you just look at this definition that the only factors are 1 and itself, it is a bit ambiguous because 1 is a factor and itself 1 is also a factor. We could naively read this as saying that 1 is a prime, but by convention 1 is not a prime.
Detailed Explanation
In this chunk, we highlight the specific case of the number 1. Although it may seem to fit the definition of a prime (having only two factors: 1 and itself), by convention, 1 is not considered a prime number. This is an essential detail to keep in mind when programming to avoid logical errors when defining primes.
Examples & Analogies
Consider 1 as the lone artist who never performs solo. Even though it may seem like they fit the criteria to be a solo artist (just like a prime), in music, a lone performance is not enough to qualify, hence the convention against defining 1 as a prime.
Listing Prime Numbers Below n
Chapter 5 of 6
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Chapter Content
What if you want to list all the prime numbers which lie below a given number n? So, we have to just check for every number from 1 to n, whether or not it is a prime. We already know how to check if something is a prime. Once again, we can write a function primesupto which takes an argument n.
Detailed Explanation
This chunk discusses how to create a function called 'primesupto'. The function takes a single argument 'n' and checks all the numbers from 1 to 'n' to see if they are prime. If a number is found to be prime, it is added to a list. Finally, once all numbers have been checked, the function returns the list of primes that are less than 'n'.
Examples & Analogies
Imagine you're searching through a crowd (numbers from 1 to n) to find all the solo singers (prime numbers). You ask each person in the crowd if they perform solo (checking primality) and take note of those who do. By the time you've reached n, you've gathered your list of solo artists (prime numbers).
Finding the First n Primes
Chapter 6 of 6
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Chapter Content
What if we change our requirements saying that we do not want primes up to n, but we want the first n primes? Now, if we want the first n primes, we do not know how many numbers to scan. We do not know where the nth prime will come.
Detailed Explanation
In this chunk, we explore the challenge of finding the first 'n' prime numbers without knowing how far we need to scan. We need to use a while loop that continues until we reach 'n' primes. We maintain a counter to track how many primes we've found and increment this counter each time we discover a prime. This requires an adaptive approach since we can't predict where the nth prime will occur.
Examples & Analogies
Think of a treasure hunt where you need to find a specified number of treasures (primes), but you have no idea how many places you need to search first. Just like you continue exploring new areas until you find all your treasures, in programming, you keep checking numbers until you've encountered enough primes.
Key Concepts
-
Factors of a Number: The factors of a number n are integers that divide n without leaving a remainder. A simple function can compute factors from 1 to n.
-
Prime Numbers: A prime number is defined as having exactly two distinct positive divisors: 1 and itself. The section provides examples of prime numbers (like 17) versus composite numbers (like 18).
-
Function Composition: Functions can call other functions, illustrated through examples of
factorsandisprime, demonstrating how to break down problems into manageable code blocks. -
Loops: The section highlights the use of 'for' when the range of numbers is known (like finding primes up to n) and 'while' when the number of iterations isn't predetermined (like finding the first n primes).
-
Code Maintainability: Emphasizes writing understandable and maintainable code through clear structure, aiding future adjustments and optimizations.
-
These principles not only foster better coding practices but also enhance the efficiency and readability of the code.
Examples & Applications
The factors of 30 are 1, 2, 3, 5, 6, 10, 15, and 30.
17 is a prime number as its only factors are 1 and 17.
Memory Aids
Interactive tools to help you remember key concepts
Rhymes
To find the primes all in a line, check the factors, one at a time!
Stories
Imagine two friends, 1 and Alex, who are prime pals only letting 1 and Alex themselves in their exclusive club.
Memory Tools
F-A-P: Factors Are Prime to remember that factors and prime definitions are closely related.
Acronyms
P.F.N
Finding Primes
Factors Necessary.
Flash Cards
Glossary
- Factors
Integers that divide a number n without a remainder.
- Prime Number
A natural number greater than 1 that has no positive divisors other than 1 and itself.
- Function
A reusable block of code that performs a specific task.
- Loop
A programming construct that repeats a set of instructions until a condition is met.
Reference links
Supplementary resources to enhance your learning experience.