Class 6 Maths Chapter 5 Notes Prime Time
Class 6 Maths Notes Chapter 5 – Class 6 Prime Time Notes
→ If a number is divisible by another, the second number is called a factor of the first. For example, 4 is a factor of 12 because 12 is divisible by 4 (12 ÷ 4 = 3).
→ Prime numbers are numbers like 2, 3, 5, 7, 11, … that have only two factors, namely 1 and themselves.
→ Composite numbers are numbers like 4, 6, 8, 9, … that have more than 2 factors, i.e., at least one factor other than 1 and themselves. For example, 8 has the factor 4, and 9 has the factor 3, so 8 and 9 are both composite.
→ Every number greater than 1 can be written as a product of prime numbers. This is called the number’s prime factorization. For example, 84 = 2 × 2 × 3 × 7.
→ There is only one way to factorize a number into primes, except for the ordering of the factors.
→ Two numbers that do not have a common factor other than 1 are said to be co-prime.
→ To check if two numbers are co-prime, we can first find their prime factorizations and check if there is a common prime factor. If there is no common prime factor, they are co-prime, and otherwise, they are not.
→ A number is a factor of another number if the prime factorization of the first number is included in the prime factorization of the second number.
Common Multiples and Common Factors Class 6 Notes
Idli-Vada Game
Children sit in a circle and play a game of numbers. One of the children starts by saying ‘1’. The second player says ‘2’, and so on. But when it is the turn of 3, 6, 9, … (multiples of 3), the player should say ‘idli’ instead of the number. When it is the turn of 5, 10, … (multiples of 5), the player should say ‘vada’ instead of the number. When a number is both a multiple of 3 and a multiple of 5, the player should say ‘idli-vada’! If a player makes any mistake, they are out.
The game continues in rounds till only one person remains. For which numbers should the players say ‘idli’ instead of saying the number? These would be 3, 6, 9, 12, 18, … and so on. For which numbers should the players say ‘vada’? These would be 5, 10, 20, … and so on. Which is the first number for which the players should say, ‘idli-vada’? It is 15, which is a multiple of 3, and also a multiple of 5. Find out other such numbers that are multiples of both 3 and 5.
Let us now play the ‘idli-vada’ game with different pairs of numbers:
a. 2 and 5,
b. 3 and 7,
c. 4 and 6.
We will say ‘idli’ for multiples of the smaller number, ‘vada’ for multiples of the larger number, and ‘idli-vada’ for common multiples. Draw a figure if the game is played up to 60.
Jump Jackpot
Jumpy and Grumpy play a game.
- 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 at 0.
- Jumpy gets the treasure if he lands on the number where Grumpy placed it.
Which jump sizes will get Jumpy to land on 24?
If he chooses 4: Jumpy lands on 4 → 8 → 12 → 16 → 20 → 24 → 28 → ……
Other successful jump sizes are 2, 3, 6, 8 and 12.
What about jump sizes 1 and 24? Yes, they also will land on 24. The numbers 1, 2, 3, 4, 6, 8, 12, 24 all divide 24 exactly. Recall that such numbers are called factors or divisors of 24. Grumpy increases the level of the game. Two treasures are kept in 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.
Grumpy has kept the treasures on 14 and 36. Jumpy chooses a jump size of 7. Will Jumpy land on both the treasures?
Starting from 0, he jumps to 7 → 14 → 21 → 28 → 35 → 42 …
We see that he landed on 14 but did not land on 36, so he does not get the treasure. What jump size should he have chosen?
The factors of 14 are 1, 2, 7, 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.
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.
The jump sizes using which both the treasures can be reached are the common factors of the two numbers where the treasures are placed.
Prime Numbers Class 6 Notes
Guna and Anshu want to pack fis (anjeer) that grow on 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 and find out the different ways.
- Guna can arrange 12 figs in a rectangular manner.
- Anshu can arrange 7 figs in a rectangular manner.
Guna has listed out these possibilities. Observe the number of rows and columns in each of the arrangements. How are they related to 12?
In the second arrangement, for example, 12 fis are arranged in two columns of 6 each or 12 = 2 × 6.
Anshu could make only one arrangement: 7 × 1 or 1 × 7. There are no other rectangular arrangements possible.
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 factor of 12. We saw that the number 12 can be arranged in a rectangle in more than one way as 12 has more than two factors. The number 7 can be arranged in only one way, as it has only two factors 1 and 7.
Numbers that have only two factors are called prime numbers or primes. Here is the fist few primes 2, 3, 5, 7, 11, 13, 17, 19. Notice that the factors of a prime number are 1 and the number itself. 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. What about 1, which has only one factor? The number 1 is neither a prime nor a composite number.
Can we list all the prime numbers from 1 to 100?
Here is an interesting way to find prime numbers. Just follow the steps given below and see what happens.
- 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, 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, and so on.
- 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, and so on.
- Step 5: Continue this process till 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. This method is called the Sieve of Eratosthenes. This procedure can be carried on for numbers greater than 100 also. Eratosthenes was a Greek mathematician who lived around 2200 years ago and developed this method of listing primes.
Guna and Anshu started wondering how this simple method can find prime numbers! Think about how this method works. Read the steps given above again and see what happens after each step is carried out.
Co-prime Numbers for Safekeeping Treasures Class 6 Notes
Which pairs are safe?
Let us go back to the treasure-finding game. This time, treasures are kept on two numbers. Jumpy gets the treasures only if he can reach both numbers with the same jump size. There is also a new rule a jump size of 1 is not allowed. Where should Grumpy place the treasures so that Jumpy cannot reach both the treasures?
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. What about 4 and 9? Jumpy cannot reach both using any jump size other than 1. So, Grumpy knows that the pair 4 and 9 are safe. Check if these pairs are safe:
a. 15 and 39
b. 4 and 15
c. 18 and 29
d. 20 and 55
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.
Example: As 15 and 39 have 3 as a common factor, they are not co-prime. But 4 and 9 are co-prime.
Prime Factorisation Class 6 Notes
Checking if two numbers are co-prime
Teacher: Are 56 and 63 co-prime?
Anshu and Guna: If they have a common factor other than 1, then they are not co-prime. Let us check.
Anshu: 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.
Guna: 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.
Guna is right, as 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.
Try another example: 80 and 63. There are many ways to factorize both numbers.
80 = 40 × 2 = 20 × 4 = 10 × 8 = 16 × 5 = ???
63 = 9 × 7 = 3 × 21 = ???
We have written ‘???’ to say that there may be more ways to factorize these numbers. But if we take any of the given factorizations, 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 there may be other ways to factorize the numbers. What this means is that we need a more systematic approach to check if two numbers are co-prime.
Prime Factorisation
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 a prime factorization 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 factorization. The idea is the same: Keep breaking the composite numbers into factors till only primes are left. The number 1 does not have any prime factorization. It is not divisible by any prime number. What is the prime factorization of a prime number like 7? It is just 7 (we cannot break it down any further).
Let us see a few more examples. By going through different ways of breaking down the number, we wrote 63 as 3 × 3 × 7 and 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 a prime factorization 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 factorization, 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 factorization!
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 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.
Prime factorization of a product of two numbers
When we find the prime factorization of a number, we first write it as a product of two factors. For example, 72 = 12 × 6. Then, we find the prime factorization of each of the factors.
In the above example, 12 = 2 × 2 × 3 and 6 = 2 × 3. Now, can you say what the prime factorization of 72 is?
The prime factorization 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 factorization of 72. Compare it with how many times it occurs in the factorizations of 12 and 6 put together. Prime factorization is of fundamental importance in the study of numbers. Let us discuss two ways in which it can be useful.
Using prime factorization to check if two numbers are co-prime
Let us again take the numbers 56 and 63. How can we check if they are co-prime? We can use the prime factorization of both numbers.
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 factorizations 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 factorization of 80 and 63? 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 1: Consider 40 and 231. Their prime factorizations 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 2: Consider 242 and 195. Their prime factorizations 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.
Using prime factorization to check if one number is divisible by another
We can say that if one number is divisible by another, the prime factorization of the second number is included in the prime factorization of the first number. We say that 48 is divisible by 12 because when we divide 48 by 12, the remainder is zero. How can we check if one number is divisible by another without carrying out long division?
Example 1: Is 168 divisible by 12? Find the prime factorizations 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 2: Is 75 divisible by 21? Find the prime factorizations 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 3: Is 42 divisible by 12? Find the prime factorizations of both:
42 = 2 × 3 × 7 and 12 = 2 × 2 × 3.
All prime factors of 12 are also prime factors of 42. However, the prime factorization of 12 is not included in the prime factorization of 42. This is because 2 occurs twice in the prime factorization of 12 but only once in the prime factorization 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 factorization of the second number is included in the prime factorization of the first number.
Divisibility Tests Class 6 Notes
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. Can you find them?
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, … Continue this sequence and observe the pattern.
Is 125 a multiple of 10? Will this number appear in the previous sequence? Why or why not?
Can you now answer if 8560 is divisible by 10?
Consider this statement: Numbers that are divisible by 10 are those that end with ‘0’. Do you agree?
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, …
What do you observe about these numbers? Do you see a pattern in the last digit?
What is the largest number less than 399 that is divisible by 5?
Is 8560 divisible by 5?
Consider this statement: Numbers that are divisible by 5 are those that end with either a ‘0’ or a ‘5’. Do you agree?
Divisibility by 2
The first few multiples of 2 are 2, 4, 6, 8, 10, 12, 14, 16, 18, 20,…
What do you observe? Do you see a pattern in the last digit?
Is 682 divisible by 2? Can we answer this without doing the long division?
Is 8560 divisible by 2? Why or why not?
Consider this statement: Numbers that are divisible by 2 are those that end with ‘0’, ‘2’, ‘4, ‘6’ or ‘8’. Do you agree?
What are all the multiples of 2 between 399 and 411?
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, ..…
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 can 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.
Divisibility by 8
Interestingly, even checking for divisibility by 8 can be simplified. Can the last two digits be used for this?
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, 8.
Do we have such simple methods for other numbers as well?
We will discuss simple methods to test divisibility by 3, 6, 7, and 9 in later classes!
Fun with Numbers Class 6 Notes
Special Numbers
There are four numbers in this box. Which number looks special to you? Why do you say so?
Look at what Guna’s classmates have to share:
- 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”.
