Definition of Factors
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Introduction to Factors
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Today, we’re going to explore factors. Can anyone tell me what a factor is?
Is it a number that can divide another number without leaving a remainder?
Exactly! Factors are numbers that divide another number evenly. For instance, if we consider the number 18, what are its factors?
Well, I think the factors of 18 are 1, 2, 3, 6, 9, and 18.
Great job! Now, remember, factors always fall between 1 and the number itself. We can compute these using a simple for loop in Python. Let’s think of a mnemonic: 'Find All Factors'. If we take a number n, we can check every number from 1 to n to see if it divides evenly. Who can suggest how we could write this in Python?
Understanding Prime Numbers
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Now, let’s move on to prime numbers. Can someone tell me what makes a number prime?
A prime number has only two factors: 1 and itself, right?
Correct! For example, the number 17 is prime because its only factors are 1 and 17. Can anyone think of a non-prime number?
The number 18 isn't prime since it has more than two factors.
Exactly! To check if a number is prime in Python, we can leverage our previously defined factors function. Can anyone remind us how we would do this?
We would check the list of factors to see if it contains only 1 and n.
Well said! It brings us to the concept of writing clean functions. Remember our motto, 'Function Overload: Break it Down!'
Finding Primes Up to n
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Now, let’s say we want to find all prime numbers below a given number n. Does anyone know how we would approach this?
We could loop through each number up to n and check if it's prime!
That’s right! By using our isprime function within a loop, we can compile all primes into a list. Can someone propose a way to implement this?
We could start with an empty list and append each prime number we find.
Excellent! This allows us to reuse our isprime function effectively. One last concept to remember: when do we use for loops and when do we use while loops?
We use for loops when we know how many iterations we need, but while loops when we don’t!
Precisely! Let’s summarize what we’ve learned today...
Using While Loops for Unknown Counts
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Let’s discuss a situation where we need to find the first n primes. How would we do this?
Since we can't estimate the upper limit, we need to use a while loop!
Exactly! We’ll keep count of the primes we find. Can someone outline the logic for that?
We can start with count = 0, check each integer, and increment the count when we find a prime.
You got it! This leads us through our while loop to continuously check for primes until we’ve collected enough - what would be our exit condition?
We stop when our count reaches n, right?
Correct! Always ensure your loop will eventually exit. Great work today, everyone!
Introduction & Overview
Read summaries of the section's main ideas at different levels of detail.
Quick Overview
Standard
The definition of factors is explored, along with the characteristics of prime numbers. The section emphasizes the importance of creating functions in programming for efficiently calculating factors and identifying prime numbers, demonstrating these concepts with example code in Python.
Detailed
In this section, we delve into the definition of factors and the characteristics of prime numbers. Factors of a number are defined as integers that divide the number evenly, and are always found in the range from 1 to n. The section includes a Python function that computes the factors of a number utilizing a for loop. To distinguish prime numbers, we establish that a prime number has only two factors: 1 and itself, exemplified by the number 17. Conversely, the number 18 has multiple factors, demonstrating that not every integer falls under the prime classification. A critical aspect mentioned is the need to prevent the number 1 from being incorrectly categorized as a prime. Additionally, the section discusses utilizing functions to check for primes and outlines methods for finding a list of all prime numbers below a given number n. The difference between using for and while loops to accomplish these tasks is highlighted, with practical examples provided. Overall, the section emphasizes the value of breaking down code into functions for clarity and ease of maintenance.
Youtube Videos
Audio Book
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Understanding Factors of a Number
Chapter 1 of 4
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Chapter Content
To round off this week, let us look at some examples to illustrate some of the concepts we have seen so far.
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
Factors are the numbers that divide another number without leaving a remainder. To find the factors of a number 'n', we can use a loop that runs from 1 to 'n'. For each integer in this range, we can check if it divides 'n' evenly—in which case it is a factor. We start with an empty list to store these factors. Each time we find a number that is a factor, we add it to our list. Finally, we return this list of factors.
Examples & Analogies
Imagine you are dividing a pizza into equal slices. The factors are the different ways you can divide the pizza without any leftover pieces. If you have a pizza with 8 slices, you can cut it into groups of 1, 2, 4, or all 8 slices. The number of slices you can make is like the factors of 8.
Identifying Prime Numbers
Chapter 2 of 4
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Chapter Content
A prime number is one which is divisible by no other number other 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 than just 1, 18. This allows us to write a very simple definition of prime based on factors which we have already seen.
Detailed Explanation
A prime number is defined as a number whose only factors are 1 and itself. For instance, the number 17 is considered prime because the only numbers that can divide it evenly are 1 and 17. In contrast, the number 18 can be divided by 1, 2, 3, 6, 9, and 18, making it a composite number, not prime. Identifying prime numbers is straightforward when we understand factors.
Examples & Analogies
Think of a club where membership is exclusive. A prime number can be compared to a club where only two members are allowed: the club president (the number itself) and the founder (the number 1). In contrast, a number like 18 is like a club with more members—everyone who has access to it can be thought of as a different factor!
The Function of Factors in Finding Primes
Chapter 3 of 4
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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 can use the factors function that we defined earlier. This function gives us all the factors of the number. We then compare the output to the list containing only 1 and the number itself (e.g., for 17, the list is [1, 17]). If both lists match, we confirm that the number is prime. This separation of tasks into functions makes our code clearer and easier to manage.
Examples & Analogies
Consider baking a cake where the recipe divides steps into smaller tasks like mixing ingredients, baking, and decorating. Just like in coding, each task (or function) works together to achieve the final delicious cake (the solution).
The Case of the Number 1
Chapter 4 of 4
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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
While the definition of a prime number seems to include 1, conventionally, 1 is not classified as a prime. This is because prime numbers are defined as having exactly two distinct factors. Since 1 only has one unique factor (which is itself), it does not meet the criteria to be a prime number.
Examples & Analogies
Imagine the number 1 as that solitary singer in a talent show who truly does not have a duet partner. While they can perform alone (being their only factor), they do not fit into the exclusive club of prime performers who each need a unique friend (themselves) while also needing someone else—this is why 1 is not prime!
Key Concepts
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Factors: Numbers that divide another number evenly.
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Prime Numbers: Natural numbers that have only two factors, 1 and itself.
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Divisors: Numbers that divide another number without a remainder.
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Functions in Python: Code blocks that can be reused throughout a program.
Examples & Applications
Factors of 18 include 1, 2, 3, 6, 9, and 18.
17 is a prime number because it can only be divided by 1 and 17.
Memory Aids
Interactive tools to help you remember key concepts
Rhymes
To find all your factors, just try ‘1 through n’, each number you check, will work time and again.
Stories
Imagine a lonely prime number walking along a path with only two friends: one being itself and the other, number 1. They can’t invite anyone else because they only belong to their small circle.
Memory Tools
Remember 'PiRC' for Prime numbers: They Only have '1' and 'C'eil(‘C’ for itself) as their friends.
Acronyms
P-Fact stands for 'Prime = Factors (1, itself)' to remember how we define primes.
Flash Cards
Glossary
- Factors
Integers that can be multiplied together to yield another integer.
- Prime Number
A natural number greater than 1 that cannot be formed by multiplying two smaller natural numbers.
- Divisor
A number that divides another without leaving a remainder.
- Function
A block of code designed to perform a particular task and can be reused within a program.
Reference links
Supplementary resources to enhance your learning experience.