Table of contents | |
Introduction | |
Common Multiples and Common Factors | |
Prime Numbers | |
Co-prime Numbers for Safe-keeping treasure | |
Prime Factorization | |
Divisibility Tests | |
Fun with Numbers |
In this chapter, we will explore the world of numbers by understanding how they can be divided and multiplied in various ways. We will learn about prime numbers, composite numbers, and how these concepts help us solve problems in mathematics. By the end of this chapter, you'll be able to identify prime numbers, understand the importance of factors and multiples, and use these concepts to solve puzzles and games.
Example: The Idli-Vada Game
Imagine you're sitting in a circle with your friends, and you're all playing a game called "Idli-Vada." The rules are simple:
The game continues until someone makes a mistake, and the last person remaining wins.
Understanding the Game: Common Multiples
Now, let's look at what’s happening in the game.
In the game, when you say "idli" or "vada," you’re recognizing the multiples of 3 and 5. When you say "idli-vada," you're identifying numbers that are shared multiples of both 3 and 5.
Example: Jump Jackpot
Now, let’s explore another game called "Jump Jackpot." Here’s how it works:
For example, if Jumpy picks 4, he lands on 4, 8, 12, 16, 20, 24... and wins the treasure! If he picks 6 or 12, he also wins because these numbers are factors of 24.
But what if there are two treasures? Let’s say one is on 14 and the other on 36. Jumpy needs to pick a number that will allow him to land on both. If he picks 7, he lands on 14 but misses 36. However, if he picks 2, he lands on both 14 and 36, winning both treasures.
Understanding the Game: Common Factors
In "Jump Jackpot," when Jumpy successfully lands on both treasures, he’s using a common factor.
The number 1 is neither a prime nor a composite number.
Example 1: The Fig Packing Game
Guna and Anshu want to pack figs from their farm. Guna decides to put 12 figs in each box, and Anshu wants to put 7 figs in each box. They start arranging the figs in rectangular shapes:
Understanding the Concepts
Example 2: Finding Prime Numbers: The Sieve of Eratosthenes
To find prime numbers between 1 and 100, you can use a method called the Sieve of Eratosthenes:
The circled numbers are primes, and the crossed-out numbers (except for 1) are composites. This method helps you quickly find all prime numbers up to 100.
Primes through the Ages
- Prime numbers are like the basic building blocks for all whole numbers. Since ancient times, over 2000 years ago, mathematicians have been trying to understand the mysteries of prime numbers, and they are still working on it today!
- One interesting question is whether there is a largest prime number or if there are an infinite number of them. A famous mathematician named Euclid found the answer to this question a long time ago, and you will learn about it in a later class.
- Here's a fun fact: The largest prime number that anyone has ever written down is so enormous that it would take about 6500 pages to write it out by hand! Because of its size, people can only keep it on a computer.
Co-prime numbers are two numbers that have no common factors other than 1. This means they can't be divided by any of the same numbers except for 1.
Imagine you have two friends, Alice and Bob, who love collecting different types of marbles. Alice has a bag with 7 marbles, and Bob has a bag with 9 marbles. They want to share their marbles in equal groups, but there’s a catch: they want to make sure no one group has the same number of marbles from both bags.
Now, let’s see if they can do this. If we try to split the marbles into groups, we notice something interesting: 7 and 9 don’t have any common number of marbles that can divide them equally, except for 1. This means Alice and Bob can’t form groups where both have the same number of marbles.
So, in this case, 7 and 9 are called co-prime numbers because they don’t share any common factors other than 1.
To make it even simpler, think of co-prime numbers as two friends who like different things and don’t have anything in common, except they both like ice cream (which is like the number 1 in this case).
Another example could be 8 and 15. These numbers are also co-prime because there’s no number other than 1 that can evenly divide both 8 and 15. Just like Alice and Bob, these numbers don’t have anything else in common!
Coprime factors are like a secret code for keeping treasures safe. If two numbers don't share any factors other than 1, they are called coprime. This means that if you place treasures on these numbers, someone trying to reach both with the same jump size (other than 1) won't be able to, making it a perfect hiding spot for your treasures!
Imagine you have a circle with pegs placed around it, like a clock face. You start by tying a piece of thread to one peg and then skipping a certain number of pegs before tying the thread to the next one. This is called the "thread-gap."
Now, let’s look at some examples to understand how this works:
15 Pegs, Thread-gap of 10:
10 Pegs, Thread-gap of 7:
14 Pegs, Thread-gap of 6:
8 Pegs, Thread-gap of 3:
What’s Happening Here?
When the number of pegs and the thread-gap are co-prime (like 10 and 7 or 8 and 3), the thread will touch every peg, creating a complete pattern. But when they’re not co-prime (like 15 and 10 or 14 and 6), the thread will skip some pegs, and you won’t get a full pattern.
Imagine you have a bunch of candies, and each candy is a different flavor. Now, you want to divide these candies into smaller groups where each group only has one type of candy.
For example, let’s say you have 56 candies. You start by grouping them into smaller piles:
Now, you’ve broken down the candies as much as possible, so you have:
These numbers—2 and 7—are prime numbers, meaning they can’t be divided into smaller groups of candies anymore. So, the prime factorization of 56 is 2 × 2 × 2 × 7.
Why Is This Important?
Prime factorization is like breaking down a number into its basic building blocks, just like separating candies into groups of the same flavor. Every number (greater than 1) can be broken down this way until you only have prime numbers left.
For example, if you want to know if two numbers, like 56 and 63, are co-prime (which means they have no common factors other than 1), you can look at their prime factorizations:
Here, you see that both numbers share the prime factor 7, so they are not co-prime. They have something in common, just like if two different candy bags had the same flavor.
Why the Order Doesn’t Matter
When you’re multiplying the numbers together to get back to the original number, the order you multiply them in doesn’t change the result.
For example:
No matter how you arrange them, multiplying the same numbers will always give you the same total. This is like saying that no matter which order you pick up the candies, you’ll still end up with the same number of candies in the end!
When we want to find the prime factorization of a number, we start by writing it as a product of two factors. For instance, we can express 72 as 12 x 6.
Understanding with an Example
Imagine you have a big bag of candies, and you want to find out what types of candies are inside by breaking it down into smaller bags. For example, let’s say you have 72 candies. You decide to first split them into two smaller bags, one with 12 candies and another with 6 candies (72 = 12 × 6).
Next, you take a look at each smaller bag:
Now, to find out the original types of candies in the big bag (which is 72), you just combine the candies from both smaller bags:
To make it easier, you can rearrange them like this:
Now, you’ve found out that the big bag of 72 candies is made up of 2’s and 3’s.
Understanding with an Example
Let’s say you have two different bags of candies, one with 56 candies and another with 63 candies. You want to check if there’s any common type of candy in both bags, meaning they aren’t co-prime.
You break down the candies in both bags:
When you compare them, you see that both bags have 7 candies. Since they share a common type of candy (which is 7), they are not co-prime.
Let’s try with two other bags, one with 80 candies and another with 63 candies:
This time, when you compare the two, you see that there are no common types of candies between them. Since they don’t share any common factors, these two numbers (80 and 63) are co-prime!
Example 1: Consider two bags, one with 40 candies and another with 231 candies.
Example 2: Consider two other bags, one with 242 candies and another with 195 candies.
Understanding with Examples
Think of numbers as baskets filled with fruits, and each type of fruit represents a prime factor. Now, you want to check if one basket (number) can completely fit into another basket without leaving anything out. If all the fruits (prime factors) of the smaller basket fit inside the bigger one, then the bigger number is divisible by the smaller number.
Example 1: Is 168 Divisible by 12?
Let’s say you have a basket with 168 fruits and another basket with 12 fruits. To check if 168 is divisible by 12, we break down the fruits (prime factors) in each basket.
Now, look at the fruits in both baskets. You can see that all the fruits in the 12-fruit basket (2 × 2 × 3) are also in the 168-fruit basket. This means you can fit the 12-fruit basket inside the 168-fruit basket without leaving anything out.
So, 168 is divisible by 12!
Example 2: Is 75 Divisible by 21?
Now, let’s take a basket with 75 fruits and another basket with 21 fruits.
When we look at the fruits, we see that the 75-fruit basket has 3 and 5, but it doesn’t have 7, which is in the 21-fruit basket. This means the smaller basket cannot fit inside the bigger one completely.
So, 75 is not divisible by 21.
Divisibility tests are like shortcuts that help you figure out if one number can be divided by another without doing long division. Instead of going through the entire process of dividing, you can use patterns or rules to quickly check.
To check if a number is divisible by 10, simply look at the last digit. If the last digit is a 0, the number is divisible by 10.
Example: Is 8560 divisible by 10?
For divisibility by 5, the rule is similar but slightly different. If the last digit is a 0 or 5, the number is divisible by 5.
Example: Is 8560 divisible by 5?
A number is divisible by 2 if its last digit is even. This means the last digit should be 0, 2, 4, 6, or 8.
Example: Is 8560 divisible by 2?
To check if a number is divisible by 4, look at the last two digits of the number. If the number formed by these two digits is divisible by 4, then the whole number is divisible by 4.
Example: Is 8536 divisible by 4?
For divisibility by 8, check the last three digits of the number. If the number formed by these three digits is divisible by 8, then the whole number is divisible by 8.
Example: Is 8560 divisible by 8?
By using these simple patterns, you can quickly check whether a number is divisible by 2, 4, 5, 8, or 10 without needing to do the full division. These rules make it much easier to handle large numbers!
Let’s take a look at how different numbers can be considered special compared to others. Here’s the first set of numbers:
9, 16, 25, 43
Each of these numbers can be seen as special in its own way, depending on how you look at them:
Now, let’s try to find out what makes numbers in the following boxes special:
Box 1: 5, 7, 12, 35
Box 2: 3, 8, 11, 24
Box 3: 27, 3, 123, 31
Box 4: 17, 27, 44, 65
In this puzzle, the goal is to fill a grid with prime numbers so that the product of each row equals the number on the right, and the product of each column equals the number below it.
Here’s how the puzzle is solved:
Puzzle Grid:
Rules to Solve the Puzzle:
Example:
By following these rules, you can solve the prime puzzle and ensure that all the conditions are met!
92 videos|348 docs|54 tests
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1. What are common multiples and how are they used in mathematics? |
2. How do you determine if two numbers are co-prime? |
3. What is prime factorization and why is it important? |
4. What are some common divisibility tests for numbers? |
5. What are prime numbers and how can they be identified? |
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