Hello my dear students! Welcome to today's mathematics lesson. I am so happy to see you all here, ready to learn something new and exciting. Today, we are going to study Chapter 5, and the title of this chapter is "Prime Time". Now, doesn't that sound interesting? Prime Time usually means the most important time, like when your favorite TV show comes on. And in mathematics, prime numbers are indeed very special and important. So let's begin our journey into the world of prime numbers together.
In this chapter, we are going to learn about many interesting concepts like common multiples, common factors, prime numbers, co-prime numbers, prime factorisation, and divisibility tests. These are all fundamental ideas in mathematics, and you will use them throughout your life. So pay attention, and don't worry if something seems difficult at first. We will take it step by step, together.
Let us start with a fun game that will help us understand the concept of multiples and common multiples.
### 5.1 Common Multiples and Common Factors
#### The Idli-Vada Game
So students, imagine you are sitting in a circle with your friends, playing a game of numbers. One child starts by saying "1", the next says "2", and so on. But here is the twist: when it is the turn of numbers like 3, 6, 9, 12, 15, 18, and so on—which are multiples of 3—you should say "idli" instead of the number. And when it is the turn of numbers like 5, 10, 15, 20, 25, and so on—which are multiples of 5—you should say "vada" instead of the number. Now, what happens when a number is both a multiple of 3 and a multiple of 5? For example, 15 is a multiple of 3 and also a multiple of 5. In that case, you should say "idli-vada"! If anyone makes a mistake, they are out of the game. The game continues until only one person remains.
Now, let me ask you a question. For which numbers should the players say "idli"? These would be 3, 6, 9, 12, 15, 18, 21, 24, and so on—all the numbers that come in the 3 times table. Similarly, for which numbers should the players say "vada"? These would be 5, 10, 15, 20, 25, 30, 35, and so on—all the numbers that come in the 5 times table.
Now here is the important question: which is the first number for which the players should say "idli-vada"? Let us think. We need a number that is both a multiple of 3 and a multiple of 5. The first such number is 15, because 15 divided by 3 gives 5, and 15 divided by 5 gives 3. So 15 is a common multiple of 3 and 5. Can you find other such numbers that are multiples of both 3 and 5? These numbers are called common multiples.
So students, let me recap what we have learned so far. A multiple of a number is what you get when you multiply that number by 1, 2, 3, 4, and so on. For example, multiples of 3 are 3, 6, 9, 12, 15, and so on. A common multiple of two numbers is a number that is a multiple of both those numbers. For 3 and 5, the common multiples are 15, 30, 45, 60, and so on.
Now, let us play this game with different pairs of numbers. Suppose we play with 2 and 5. We will say "idli" for multiples of 2 and "vada" for multiples of 5. The common multiples will be "idli-vada". Similarly, we can play with 3 and 7, or 4 and 6.
Now, here is an interesting observation that someone made while playing this game: "Yesterday, we played this game with two numbers. We ended up saying just 'idli' or 'idli-vada' and nobody said just 'vada'! One of the numbers was 4." So students, can you think about which number could be the other number? The options are 2, 3, 5, 8, and 10. Think about it carefully. If one number is 4, and when we play the game, nobody says just "vada", that means every multiple of the other number is also a multiple of 4. In other words, the other number should be a factor of 4. The factors of 4 are 1, 2, and 4. Among the options, 2 is a factor of 4. But if the other number is 2, then multiples of 2 are 2, 4, 6, 8, 10, 12, and so on. Now, 4 is a multiple of both 2 and 4, so we would say "idli-vada" for 4. But what about 6? 6 is a multiple of 2 but not a multiple of 4, so we would say just "idli". What about 8? 8 is a multiple of both 2 and 4, so we would say "idli-vada". What about 10? 10 is a multiple of 2 but not a multiple of 4, so we would say just "idli". So if the other number is 2, we would sometimes say just "idli" and sometimes "idli-vada", but never just "vada". That matches what the person said! So the other number is 2. Good, isn't it?
Now, let us move on to another game that will help us understand factors.
#### Jump Jackpot Game
So students, let us imagine another game. There are two players: Grumpy and Jumpy. Grumpy places a treasure on some number. For example, he may place it on 24. Jumpy chooses a jump size. If he chooses 4, then he has to jump only on multiples of 4, starting from 0. So he would land on 4, then 8, then 12, then 16, then 20, then 24, then 28, and so on. Jumpy gets the treasure if he lands on the number where Grumpy placed it. Since Grumpy placed the treasure on 24, and Jumpy lands on 24 when he jumps by 4, he gets the treasure!
Now, which jump sizes will get Jumpy to land on 24? Let us think. If he chooses a jump size of 1, he will land on every number: 1, 2, 3, 4, 5, and so on, so he will definitely land on 24. If he chooses 2, he will land on 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, and so on. If he chooses 3, he will land on 3, 6, 9, 12, 15, 18, 21, 24, and so on. If he chooses 4, he will land on 4, 8, 12, 16, 20, 24, and so on. If he chooses 6, he will land on 6, 12, 18, 24, and so on. If he chooses 8, he will land on 8, 16, 24, and so on. If he chooses 12, he will land on 12, 24, and so on. And if he chooses 24, he will land on 24 directly.
So the jump sizes that will get Jumpy to land on 24 are 1, 2, 3, 4, 6, 8, 12, and 24. Do you notice something about these numbers? They are all numbers that divide 24 exactly. For example, 2 divides 24 exactly because 24 divided by 2 is 12 with no remainder. Similarly, 3 divides 24 exactly because 24 divided by 3 is 8 with no remainder. These numbers are called factors or divisors of 24. So students, the factors of a number are the numbers that divide it exactly, leaving no remainder.
Now, let us make the game more interesting. Grumpy increases the level of the game. Two treasures are kept on two different numbers. Jumpy has to choose a jump size and stick to it. Jumpy gets the treasures only if he lands on both the numbers with the chosen jump size. As before, Jumpy starts at 0.
So students, suppose Grumpy has kept the treasures on 14 and 36. And Jumpy chooses a jump size of 7. Will Jumpy land on both the treasures? Starting from 0, he jumps to 7, then 14, then 21, then 28, then 35, then 42, and so on. We see that he landed on 14, but he did not land on 36. So he does not get the treasure. What jump size should he have chosen?
Let us think about this. The factors of 14 are: 1, 2, 7, and 14. So these jump sizes will land on 14. The factors of 36 are: 1, 2, 3, 4, 6, 9, 12, 18, and 36. These jump sizes will land on 36. Now, which jump sizes are common to both lists? The numbers 1 and 2 appear in both lists. So the jump sizes of 1 or 2 will land on both 14 and 36. Notice that 1 and 2 are the common factors of 14 and 36.
So students, let me explain this clearly. The factors of a number are the numbers that divide it exactly. The common factors of two numbers are the factors that are common to both numbers. In this case, the common factors of 14 and 36 are 1 and 2. These are the jump sizes using which both the treasures can be reached.
Now, let me ask you a question. What jump size can reach both 15 and 30? There are multiple jump sizes possible. Can you find them all? The factors of 15 are 1, 3, 5, and 15. The factors of 30 are 1, 2, 3, 5, 6, 10, 15, and 30. The common factors are 1, 3, 5, and 15. So the jump sizes that can reach both 15 and 30 are 1, 3, 5, and 15.
Now, let me introduce you to some special numbers. There is something called a perfect number. A number for which the sum of all its factors is equal to twice the number is called a perfect number. For example, the number 28 is a perfect number. Its factors are 1, 2, 4, 7, 14, and 28. If we add them up, we get 1 + 2 + 4 + 7 + 14 + 28 = 56, which is twice 28. So 28 is a perfect number. Can you find a perfect number between 1 and 10? Let us check. The factors of 6 are 1, 2, 3, and 6. Their sum is 1 + 2 + 3 + 6 = 12, which is twice 6. So 6 is a perfect number! Good.
Now, let us look at a table of numbers. In this table, some numbers are shaded, some are circled, and some are both shaded and circled. Let us see what we can observe.
First, is there anything common among the shaded numbers? If the shaded numbers are multiples of 3, then yes, they are all multiples of 3. Second, is there anything common among the circled numbers? If the circled numbers are multiples of 4, then yes, they are all multiples of 4. Third, which numbers are both shaded and circled? These would be numbers that are multiples of both 3 and 4, which means they are multiples of 12. These numbers are called common multiples of 3 and 4.
So students, let me recap what we have learned in this section. We learned about multiples, which are numbers obtained by multiplying a given number by 1, 2, 3, and so on. We learned about factors or divisors, which are numbers that divide a given number exactly. We learned about common multiples, which are multiples of two or more numbers. We learned about common factors, which are factors that are common to two or more numbers. And we learned about perfect numbers, which are numbers whose factors sum to twice the number.
Now, let us move on to a very important concept: prime numbers.
### 5.2 Prime Numbers
So students, let us imagine that Guna and Anshu want to pack figs, also called anjeer, that grow in their farm. Guna wants to put 12 figs in each box, and Anshu wants to put 7 figs in each box. How many arrangements are possible?
Think about this. Guna wants to arrange 12 figs in a rectangular manner. That means he wants to arrange them in rows and columns, like a rectangle. For example, he could put 12 figs in one row and 1 column, which is 12 × 1. Or he could put 6 figs in two rows, which is 6 × 2. Or he could put 4 figs in three rows, which is 4 × 3. Or he could put 3 figs in four rows, which is 3 × 4. Or he could put 2 figs in six rows, which is 2 × 6. Or he could put 1 fig in twelve rows, which is 1 × 12. So there are multiple ways to arrange 12 figs in a rectangle.
Now, what about Anshu? He wants to arrange 7 figs in a rectangular manner. Can you find all the ways? He could put 7 figs in one row and 1 column, which is 7 × 1. Or he could put 1 fig in seven rows, which is 1 × 7. But are there any other ways? No, because 7 is a prime number, and it can only be divided by 1 and 7. So there are only two ways to arrange 7 figs in a rectangle.
Now, let us think about the relationship between the number of rows and columns and the number of figs. In each of Guna's arrangements, multiplying the number of rows by the number of columns gives the number 12. So, the number of rows or columns are factors of 12. We saw that the number 12 can be arranged in a rectangle in more than one way because it has more than two factors. The number 7 can be arranged in only one way because it has only two factors: 1 and 7.
So students, here is the key idea. Numbers that have only two factors are called prime numbers or primes. The two factors are 1 and the number itself. For example, the factors of 7 are 1 and 7, so 7 is a prime number. The factors of 12 are 1, 2, 3, 4, 6, and 12, so 12 is not a prime number. Here are the first few prime numbers: 2, 3, 5, 7, 11, 13, 17, 19. Notice that 2 is the smallest prime number, and it is also the only even prime number. All other even numbers have at least 3 factors: 1, 2, and the number itself, so they are not prime.
Now, what about numbers that have more than two factors? They are called composite numbers. The first few composite numbers are: 4, 6, 8, 9, 10, 12, 14, 15, 16, 18, 20. Notice that 4 has factors 1, 2, and 4. 6 has factors 1, 2, 3, and 6. And so on.
What about the number 1? It has only one factor: 1 itself. So the number 1 is neither a prime nor a composite number. It is a special number.
Now, let me ask you a question. How many prime numbers are there from 21 to 30? Let us list the numbers from 21 to 30: 21, 22, 23, 24, 25, 26, 27, 28, 29, 30. Among these, 23 and 29 are prime numbers because they have only two factors: 1 and themselves. So there are 2 prime numbers from 21 to 30. How many composite numbers are there from 21 to 30? The remaining 8 numbers are composite. So there are 8 composite numbers from 21 to 30.
Now, here is an interesting question. Can we list all the prime numbers from 1 to 100? There is an interesting way to find prime numbers, and it is called the Sieve of Eratosthenes. Let me explain how it works.
Step 1: Cross out 1 because it is neither prime nor composite.
Step 2: Circle 2, and then cross out all multiples of 2 after that, i.e., 4, 6, 8, 10, and so on.
Step 3: You will find that the next uncrossed number is 3. Circle 3 and then cross out all the multiples of 3 after that, i.e., 6, 9, 12, 15, and so on. Notice that some of these numbers might already be crossed out, but that's okay.
Step 4: The next uncrossed number is 5. Circle 5 and then cross out all the multiples of 5 after that, i.e., 10, 15, 20, 25, and so on.
Step 5: Continue this process until all the numbers in the list are either circled or crossed out.
All the circled numbers are prime numbers. All the crossed out numbers, other than 1, are composite numbers.
This method is called the Sieve of Eratosthenes. Eratosthenes was a Greek mathematician who lived around 2200 years ago and developed this method of listing primes. Isn't that amazing? More than 2000 years ago, people were already studying prime numbers!
Now, let me explain why this method works. When we circle a number and cross out its multiples, we are essentially saying that this number is prime because it has not been crossed out by any smaller number. Once we have processed all the numbers up to 10, we have effectively identified all the primes up to 100. This is because any composite number has a prime factor that is less than or equal to its square root. For 100, the square root is 10, so we only need to check primes up to 10.
So students, let me recap what we have learned in this section. We learned that prime numbers are numbers that have exactly two factors: 1 and the number itself. We learned that composite numbers are numbers that have more than two factors. We learned that 1 is neither prime nor composite. We learned about the Sieve of Eratosthenes, which is a method to find all prime numbers up to a given limit.
Now, let me tell you some interesting facts about prime numbers. Prime numbers are the building blocks of all whole numbers. Starting from the time of the Greek civilization more than 2000 years ago to this day, mathematicians are still struggling to uncover their secrets! One question that people have asked is: is there a largest prime number? Or does the list of prime numbers go on without an end? A mathematician named Euclid found the answer, and you will learn about it in a later class. The answer is that there is no largest prime number—the list goes on forever!
Here is a fun fact: the largest prime number that anyone has "written down" is so large that it would take around 6500 pages to write it! So they could only write it on a computer!
Now, let us move on to another important concept: co-prime numbers.
### 5.3 Co-prime Numbers for Safekeeping Treasures
So students, let us go back to the treasure finding game. This time, treasures are kept on two numbers. Jumpy gets the treasures only if he is able to reach both the numbers with the same jump size. There is also a new rule: a jump size of 1 is not allowed.
Now, the question is: where should Grumpy place the treasures so that Jumpy cannot reach both the treasures? In other words, Grumpy wants to place the treasures on numbers that have no common factor other than 1. Such pairs are called safe pairs.
Let us check some examples. Will placing the treasure on 12 and 26 work? No! If the jump size is chosen to be 2, then Jumpy will reach both 12 and 26. So this is not a safe pair.
What about 4 and 9? Let us check. The factors of 4 are 1, 2, and 4. The factors of 9 are 1, 3, and 9. The only common factor is 1. So Jumpy cannot reach both using any jump size other than 1, and 1 is not allowed. So Grumpy knows that the pair 4 and 9 is safe.
Now, let me ask you to check if these pairs are safe: 15 and 39, 4 and 15, 18 and 29, 20 and 55.
For 15 and 39, the factors of 15 are 1, 3, 5, and 15. The factors of 39 are 1, 3, 13, and 39. The common factors are 1 and 3. So they are not co-prime, and this is not a safe pair.
For 4 and 15, we already saw that the only common factor is 1. So they are co-prime, and this is a safe pair.
For 18 and 29, the factors of 18 are 1, 2, 3, 6, 9, and 18. The factors of 29 are 1 and 29. The only common factor is 1. So they are co-prime, and this is a safe pair.
For 20 and 55, the factors of 20 are 1, 2, 4, 5, 10, and 20. The factors of 55 are 1, 5, 11, and 55. The common factors are 1 and 5. So they are not co-prime, and this is not a safe pair.
So students, what is special about safe pairs? They don't have any common factor other than 1. Two numbers are said to be co-prime to each other if they have no common factor other than 1.
For example, as we saw, 15 and 39 have 3 as a common factor, so they are not co-prime. But 4 and 9 are co-prime because their only common factor is 1.
Now, let me ask you to identify which of these pairs are co-prime: 18 and 35, 15 and 37, 30 and 415, 17 and 69, 81 and 18.
For 18 and 35, the factors of 18 are 1, 2, 3, 6, 9, and 18. The factors of 35 are 1, 5, 7, and 35. The only common factor is 1. So they are co-prime.
For 15 and 37, the factors of 15 are 1, 3, 5, and 15. The factors of 37 are 1 and 37. The only common factor is 1. So they are co-prime.
For 30 and 415, the factors of 30 are 1, 2, 3, 5, 6, 10, 15, and 30. The factors of 415 are 1, 5, 83, and 415. The common factors are 1 and 5. So they are not co-prime.
For 17 and 69, the factors of 17 are 1 and 17. The factors of 69 are 1, 3, 23, and 69. The only common factor is 1. So they are co-prime.
For 81 and 18, the factors of 81 are 1, 3, 9, 27, and 81. The factors of 18 are 1, 2, 3, 6, 9, and 18. The common factors are 1, 3, and 9. So they are not co-prime.
Now, while playing the 'idli-vada' game with different number pairs, Anshu observed something interesting! Sometimes the first common multiple was the same as the product of the two numbers. At other times, the first common multiple was less than the product of the two numbers. Let me explain this.
If two numbers are co-prime, then their first common multiple is always their product. For example, 3 and 5 are co-prime. Their product is 15, and 15 is also their first common multiple. Similarly, 4 and 9 are co-prime. Their product is 36, and 36 is also their first common multiple.
But if two numbers are not co-prime, then their first common multiple is less than their product. For example, 3 and 6 are not co-prime because they have a common factor of 3. Their product is 18, but their first common multiple is 6, which is less than 18. Similarly, 4 and 6 are not co-prime because they have a common factor of 2. Their product is 24, but their first common multiple is 12, which is less than 24.
So students, here is an important observation: whenever the two numbers are co-prime, their first common multiple is equal to their product. When they are not co-prime, their first common multiple is less than their product.
Now, let me tell you about something called co-prime art. Observe the following thread art. The first diagram has 12 pegs, and the thread is tied to every fourth peg. We say that the thread-gap is 4. The second diagram has 13 pegs, and the thread-gap is 3. What about the other diagrams? In some diagrams, the thread is tied to every peg. In some, it is not. This is related to the two numbers—the number of pegs and the thread-gap—being co-prime. If the number of pegs and the thread-gap are co-prime, then the thread will eventually be tied to every peg. If they are not co-prime, then there will be some pegs that the thread never reaches.
So students, let me recap what we have learned in this section. We learned that two numbers are co-prime if they have no common factor other than 1. We learned that if two numbers are co-prime, their first common multiple is equal to their product. We learned about co-prime art and how it relates to co-prime numbers.
Now, let us move on to a very important concept: prime factorisation.
### 5.4 Prime Factorisation
So students, we have learned how to check if two numbers are co-prime by finding their common factors. But as we saw earlier, sometimes we might miss some factors if we don't list all of them. For example, let us check if 56 and 63 are co-prime.
Anshu said: "I can write 56 = 14 × 4 and 63 = 21 × 3. So, 14 and 4 are factors of 56. Further, 21 and 3 are factors of 63. So, there are no common factors. The numbers are co-prime."
But Guna said: "Hold on. I can also write 56 = 7 × 8 and 63 = 9 × 7. We see that 7 is a factor of both numbers, so they are not co-prime."
Clearly Guna is right, because 7 is a common factor. But where did Anshu go wrong? Writing 56 = 14 × 4 tells us that 14 and 4 are both factors of 56, but it does not tell all the factors of 56. The same holds for the factors of 63. So we need a more systematic approach to check if two numbers are co-prime.
Let us try another example: 80 and 63. There are many ways to factorise both numbers.
80 = 40 × 2 = 20 × 4 = 10 × 8 = 16 × 5 63 = 9 × 7 = 3 × 21
If we take any of the given factorisations, for example, 80 = 16 × 5 and 63 = 9 × 7, then there are no common factors. Can we conclude that 80 and 63 are co-prime? As Anshu's mistake above shows, we cannot conclude that because there may be other ways to factorise the numbers. So we need a more systematic approach, and that is prime factorisation.
Now, let me explain what prime factorisation is. Take a number such as 56. It is composite, as we saw that it can be written as 56 = 4 × 14. So both 4 and 14 are factors of 56. Now take one of these, say 14. It is also composite and can be written as 14 = 2 × 7. Therefore, 56 = 4 × 2 × 7. Now, 4 is composite and can be written as 4 = 2 × 2. Therefore, 56 = 2 × 2 × 2 × 7. All the factors appearing here, 2 and 7, are prime numbers. So we cannot divide them further.
In conclusion, we have written 56 as a product of prime numbers. This is called prime factorisation of 56. The individual factors are called prime factors. For example, the prime factors of 56 are 2 and 7.
Every number greater than 1 has a prime factorisation. The idea is the same: keep breaking the composite numbers into factors until only primes are left.
The number 1 does not have any prime factorisation. It is not divisible by any prime number.
What is the prime factorisation of a prime number like 7? It is just 7, because we cannot break it down any further.
Now, let us see a few more examples. We can write 63 as 3 × 3 × 7 and also as 3 × 7 × 3. Are they different? Not really! The same prime numbers 3 and 7 occur in both cases. Further, 3 appears two times in both, and 7 appears once.
Here, you see four different ways to get prime factorisation of 36. Observe that in all four cases, we get two 2s and two 3s. Multiply back to see that you get 36 in all four cases.
For any number, it is a remarkable fact that there is only one prime factorisation, except that the prime factors may come in different orders. As we explain below, the order is not important. However, as we saw in these examples, there are many ways to arrive at the prime factorisation!
Does the order matter? Using this diagram, can you explain why 30 = 2 × 3 × 5, no matter which way you multiply 2, 3, and 5? When multiplying numbers, we can do so in any order. The end result is the same. That is why, when two 2s and two 3s are multiplied in any order, we get 36. In a later class, we shall study this under the names of commutativity and associativity of multiplication. Thus, the order does not matter. Usually we write the prime numbers in increasing order. For example, 225 = 3 × 3 × 5 × 5 or 30 = 2 × 3 × 5.
Now, let us talk about prime factorisation of a product of two numbers. When we find the prime factorisation of a number, we first write it as a product of two factors. For example, 72 = 12 × 6. Then, we find the prime factorisation of each of the factors. In the above example, 12 = 2 × 2 × 3 and 6 = 2 × 3. Now, can you say what the prime factorisation of 72 is?
The prime factorisation of the original number is obtained by putting these together. 72 = 2 × 2 × 3 × 2 × 3. We can also write this as 2 × 2 × 2 × 3 × 3. Multiply and check that you get 72 back! Observe how many times each prime factor occurs in the factorisation of 72. Compare it with how many times it occurs in the factorisations of 12 and 6 put together.
Now, let me explain how prime factorisation can be useful in checking if two numbers are co-prime.
Let us take the numbers 56 and 63 again. How can we check if they are co-prime using prime factorisation?
56 = 2 × 2 × 2 × 7 and 63 = 3 × 3 × 7.
Now, we see that 7 is a prime factor of 56 as well as 63. Therefore, 56 and 63 are not co-prime.
What about 80 and 63? Their prime factorisations are as follows:
80 = 2 × 2 × 2 × 2 × 5 and 63 = 3 × 3 × 7.
There are no common prime factors. Can we conclude that they are co-prime? Suppose they have a common factor that is composite. Would the prime factors of this composite common factor appear in the prime factorisation of 80 and 63? If there is a composite common factor, then it must be made up of prime factors. But since there are no common prime factors, there cannot be any composite common factor either. Therefore, we can say that if there are no common prime factors, then the two numbers are co-prime.
Let us see some examples.
Example: Consider 40 and 231. Their prime factorisations are as follows:
40 = 2 × 2 × 2 × 5 and 231 = 3 × 7 × 11.
We see that there are no common primes that divide both 40 and 231. Indeed, the prime factors of 40 are 2 and 5, while the prime factors of 231 are 3, 7, and 11. Therefore, 40 and 231 are co-prime!
Example: Consider 242 and 195. Their prime factorisations are as follows:
242 = 2 × 11 × 11 and 195 = 3 × 5 × 13.
The prime factors of 242 are 2 and 11. The prime factors of 195 are 3, 5, and 13. There are no common prime factors. Therefore, 242 and 195 are co-prime.
Now, let me explain how prime factorisation can be used to check if one number is divisible by another.
We can say that if one number is divisible by another, the prime factorisation of the second number is included in the prime factorisation of the first number.
Example: Is 168 divisible by 12? Find the prime factorisations of both:
168 = 2 × 2 × 2 × 3 × 7 and 12 = 2 × 2 × 3.
Since we can multiply in any order, now it is clear that 168 = 2 × 2 × 3 × 2 × 7 = 12 × 14. Therefore, 168 is divisible by 12.
Example: Is 75 divisible by 21? Find the prime factorisations of both:
75 = 3 × 5 × 5 and 21 = 3 × 7.
As we saw in the discussion above, if 75 was a multiple of 21, then all prime factors of 21 would also be prime factors of 75. However, 7 is a prime factor of 21 but not a prime factor of 75. Therefore, 75 is not divisible by 21.
Example: Is 42 divisible by 12? Find the prime factorisations of both:
42 = 2 × 3 × 7 and 12 = 2 × 2 × 3.
All prime factors of 12 are also prime factors of 42. But the prime factorisation of 12 is not included in the prime factorisation of 42. This is because 2 occurs twice in the prime factorisation of 12 but only once in the prime factorisation of 42. This means that 42 is not divisible by 12.
We can say that if one number is divisible by another, then the prime factorisation of the second number is included in the prime factorisation of the first number.
So students, let me recap what we have learned in this section. We learned about prime factorisation, which is writing a number as a product of prime numbers. We learned that every number greater than 1 has a unique prime factorisation (except for the order of the factors). We learned that we can use prime factorisation to check if two numbers are co-prime by checking if they have any common prime factors. We learned that we can use prime factorisation to check if one number is divisible by another by checking if the prime factorisation of the second number is included in the prime factorisation of the first number.
Now, let us move on to the next section: divisibility tests.
### 5.5 Divisibility Tests
So students, so far we have been finding factors of numbers in different contexts, including to determine if a number is prime or not, or if a given pair of numbers is co-prime or not. It is easy to find factors of small numbers. How do we find factors of a large number?
Let us take 8560. Does it have any factors from 2 to 10 (2, 3, 4, 5, ..., 9, 10)? It is easy to check if some of these numbers are factors or not without doing long division. Let us see how.
#### Divisibility by 10
Let us take 10. Is 8560 divisible by 10? This is another way of asking if 10 is a factor of 8560.
For this, we can look at the pattern in the multiples of 10. The first few multiples of 10 are: 10, 20, 30, 40, 50, 60, 70, 80, 90, 100, and so on. What do you notice? All these numbers end with 0. So, numbers that are divisible by 10 are those that end with 0. Is 125 a multiple of 10? No, because 125 ends with 5, not 0. So 125 is not a multiple of 10. Can you now answer if 8560 is divisible by 10? Yes, because 8560 ends with 0. So 8560 is divisible by 10.
#### Divisibility by 5
The number 5 is another number whose divisibility can easily be checked. How do we do it? Explore by listing down the multiples: 5, 10, 15, 20, 25, 30, 35, 40, 45, 50, and so on. What do you observe about these numbers? Do you see a pattern in the last digit? All these numbers end with either 0 or 5. So, numbers that are divisible by 5 are those that end with either 0 or 5. What is the largest number less than 399 that is divisible by 5? It is 395. Is 8560 divisible by 5? Yes, because 8560 ends with 0.
#### Divisibility by 2
The first few multiples of 2 are 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, and so on. What do you observe? Do you see a pattern in the last digit? All these numbers end with 0, 2, 4, 6, or 8. These are called even numbers. So, numbers that are divisible by 2 are those that end with 0, 2, 4, 6, or 8. Is 682 divisible by 2? Yes, because 682 ends with 2. Is 8560 divisible by 2? Yes, because 8560 ends with 0.
#### Divisibility by 4
Checking if a number is divisible by 4 can also be done easily! Look at its multiples: 4, 8, 12, 16, 20, 24, 28, 32, 36, 40, and so on. Are you able to observe any patterns that can be used? The multiples of 10, 5, and 2 have a pattern in their last digits which we are able to use to check for divisibility. Similarly, can we check if a number is divisible by 4 by looking at the last digit?
It does not work! Look at 12 and 22. They have the same last digit, but 12 is a multiple of 4 while 22 is not. Similarly, 14 and 24 have the same last digit, but 14 is not a multiple of 4 while 24 is. Similarly, 16 and 26, or 18 and 28. What this means is that by looking at the last digit, we cannot tell whether a number is a multiple of 4.
Can we answer the question by looking at more digits? Make a list of multiples of 4 between 1 and 200 and search for a pattern. Find numbers between 330 and 340 that are divisible by 4. Also, find numbers between 1730 and 1740, and 2030 and 2040, that are divisible by 4. What do you observe?
You might notice that for divisibility by 4, we need to look at the last two digits of the number. If the number formed by the last two digits is divisible by 4, then the original number is divisible by 4. And conversely, if the original number is divisible by 4, then the number formed by the last two digits is divisible by 4.
For example, is 8536 divisible by 4? The last two digits are 36. 36 divided by 4 is 9, so yes, 8536 is divisible by 4.
So students, here are the statements about divisibility by 4:
1. Only the last two digits matter when deciding if a given number is divisible by 4. 2. If the number formed by the last two digits is divisible by 4, then the original number is divisible by 4. 3. If the original number is divisible by 4, then the number formed by the last two digits is divisible by 4.
All three statements are true.
#### Divisibility by 8
Interestingly, even checking for divisibility by 8 can be simplified. Can the last two digits be used for this? Let us see.
Find numbers between 120 and 140 that are divisible by 8. Also find numbers between 1120 and 1140, and 3120 and 3140, that are divisible by 8. What do you observe?
You might notice that for divisibility by 8, we need to look at the last three digits of the number. If the number formed by the last three digits is divisible by 8, then the original number is divisible by 8. And conversely, if the original number is divisible by 8, then the number formed by the last three digits is divisible by 8.
For example, is 8560 divisible by 8? The last three digits are 560. 560 divided by 8 is 70, so yes, 8560 is divisible by 8.
So students, here are the statements about divisibility by 8:
1. Only the last three digits matter when deciding if a given number is divisible by 8. 2. If the number formed by the last three digits is divisible by 8, then the original number is divisible by 8. 3. If the original number is divisible by 8, then the number formed by the last three digits is divisible by 8.
All three statements are true.
So students, we have seen that long division is not always needed to check if a number is a factor or not. We have made use of certain observations to come up with simple methods for 10, 5, 2, 4, and 8. We will discuss simple methods to test divisibility by 3, 6, 7, and 9 in later classes!
Now, let me recap what we have learned in this section. We learned divisibility tests for 10, 5, 2, 4, and 8. For divisibility by 10, we check if the number ends with 0. For divisibility by 5, we check if the number ends with 0 or 5. For divisibility by 2, we check if the number ends with 0, 2, 4, 6, or 8. For divisibility by 4, we check if the number formed by the last two digits is divisible by 4. For divisibility by 8, we check if the number formed by the last three digits is divisible by 8.
### 5.6 Fun with Numbers
So students, now let us have some fun with numbers. There are four numbers in this box: 9, 16, 25, and 43. Which number looks special to you? Why do you say so?
Let me tell you what different students said:
Karnawati says, "9 is special because it is a single-digit number whereas all the other numbers are 2-digit numbers."
Gurupreet says, "9 is special because it is the only number that is a multiple of 3."
Murugan says, "16 is special because it is the only even number and also the only multiple of 4."
Gopika says, "25 is special as it is the only multiple of 5."
Yadnyikee says, "43 is special because it is the only prime number."
Radha says, "43 is special because it is the only number that is not a square."
So students, you can see that there are many ways to look at numbers and find what makes them special. This is the beauty of mathematics!
Now, let me tell you about a prime puzzle. In this puzzle, you have to fill the grid with prime numbers only so that the product of each row is the number to the right of the row, and the product of each column is the number below the column. This is a fun way to practice your understanding of prime numbers and multiplication.
So students, we have come to the end of this chapter. Let me now give you a complete summary of everything we have learned.
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## Complete Summary
So students, in this chapter, we learned about many important concepts in mathematics. Let me summarize everything we have learned.
First, we learned about common multiples and common factors through the Idli-Vada game and the Jump Jackpot game. We learned that a multiple of a number is what you get when you multiply that number by 1, 2, 3, and so on. We learned that a factor or divisor of a number is a number that divides it exactly without leaving a remainder. We learned that a common multiple of two numbers is a number that is a multiple of both, and a common factor of two numbers is a number that is a factor of both. We also learned about perfect numbers, which are numbers whose factors sum to twice the number. The number 6 is a perfect number between 1 and 10.
Second, we learned about prime numbers. We learned that prime numbers are numbers that have exactly two factors: 1 and the number itself. We learned that composite numbers are numbers that have more than two factors. We learned that 1 is neither prime nor composite. We learned about the Sieve of Eratosthenes, which is a method to find all prime numbers up to a given limit. We learned that 2 is the only even prime number. We learned that prime numbers are the building blocks of all whole numbers, and there is no largest prime number.
Third, we learned about co-prime numbers. We learned that two numbers are co-prime if they have no common factor other than 1. We learned that if two numbers are co-prime, their first common multiple is equal to their product. We learned about co-prime art and how it relates to co-prime numbers.
Fourth, we learned about prime factorisation. We learned that prime factorisation is writing a number as a product of prime numbers. We learned that every number greater than 1 has a unique prime factorisation (except for the order of the factors). We learned that we can use prime factorisation to check if two numbers are co-prime by checking if they have any common prime factors. We learned that we can use prime factorisation to check if one number is divisible by another by checking if the prime factorisation of the second number is included in the prime factorisation of the first number.
Fifth, we learned about divisibility tests. We learned simple tests to check if a number is divisible by 10, 5, 2, 4, and 8. For divisibility by 10, we check if the number ends with 0. For divisibility by 5, we check if the number ends with 0 or 5. For divisibility by 2, we check if the number ends with 0, 2, 4, 6, or 8. For divisibility by 4, we check if the number formed by the last two digits is divisible by 4. For divisibility by 8, we check if the number formed by the last three digits is divisible by 8.
Sixth, we had some fun with numbers. We learned that there are many ways to look at numbers and find what makes them special. We also learned about a prime puzzle that involves filling a grid with prime numbers.
So students, this is everything we have learned in Chapter 5: Prime Time. I hope you enjoyed this lesson and understood all the concepts. Remember, mathematics is all about understanding concepts and practicing them. So keep practicing, and you will become great at math!
Thank you for listening attentively. See you in the next lesson!