PLAYING WITH NUMBERS| CHAPTER 3| EX 3.5| GRADE 6| INSIGHTFUL MATHS
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00:00Hi everyone, welcome to insightful maths. Today's session is about prime factorization.
00:06We will be doing this topic using exercise 3.5 playing with numbers grade 6.
00:13Before starting the session, if you have not liked and subscribed to my channel, please like and subscribe.
00:20So the session starts with the first topic which is prime factorization.
00:27In prime factorization, what we do? Any number should be written as the product of its prime factors.
00:35Now what does it mean? If I have a number 12, so I can write this number 12 as 2 multiplied by 6.
00:43But 2 is a prime number, 6 is not. It's a composite number. So this is not the prime factorization.
00:51So how I am going to do it? Start the repeated division of this number 12.
00:57We can do this directly as well or otherwise if you feel this method useful, you can do it in this manner.
01:032 x 6 is 12, 2 x 3 is 6 and 3 x 1 is 3. Right? We have done the repeated division.
01:14Please ensure one thing that we never include this 1 in the prime factorization.
01:20Why is that so? Because 1 is not a prime number. So we have to write this 12 as the product of 2 x 2 x 3.
01:32This is the prime factorization of 12. Let's do one more example.
01:37If we have the number 36, you can see here prime factorizations are being done.
01:43I am doing the repeated division for you to understand how you have to prime factorize any given numbers.
01:50Now 36 being a prime number, it is divisible by 2. So 36 x 2, that's 18.
01:5818 x 2 is 9, 9 x 3 is 3 and 3 x 3 is 1. We are not going to include this 1 as it's not a prime number.
02:12So out of the given options, which is the correct answer for prime factorization?
02:17Have a look. These all products will give you 36. But 18 is composite, 12, 9, 4.
02:27These are the composite numbers. So it is not a prime factorization.
02:31So this is the correct answer. Have a look. 2, 2, 3 and 3.
02:37So you have to write the final answer like this. 36 is the product of 2, 2, 3 and 3.
02:46That's the answer for prime factorization.
02:49Moving on to the next topic. Making a factor tree of the given number.
02:55Please understand if I have a number A. Okay, any random number.
03:00If I am branching this number, so the product of these two numbers should give you A.
03:07What does that mean? If I am making the factor tree, let's say of number 36.
03:13Advisable, break it in such a manner that at least one number should be the prime number.
03:19So I know that this is an even number. So 1 prime number 2.
03:24Let us circle every time we are getting the prime number.
03:28It will make your task easier when you write your final answer.
03:31So 2 into which number gives you 36? That is 18. We'll write 18 here.
03:38Once again, 18 can be broken in two parts. One number, even number 2, prime number I am again taking.
03:472 into 9 is 18. 9 is composite here, which can again be broken as 3 multiplied by 3.
03:56Now how many factors have we got? 1, 2, 3, 4. Is it not looking like a tree?
04:02It's looking like a tree. That's why the name here given is factor tree method.
04:08So again, we have bought the matrix of 36 as this one 2.
04:13Again, one more 2. Then we got 3 and 3. That's your final answer using factor tree method.
04:23So after this, these are the basic topics we'll be using in this particular exercise.
04:28You will be understanding it better once we are going to do the textbook questions.
04:33So starting with question number 1 of exercise 3.5, we have the starting number as 60.
04:40It is split in two parts. Check once again. 6 into 10. It is giving you 60. Easy enough.
04:47Now the product of which two numbers will give you 6? 2 multiplied by 3 gives you 6. That's the answer.
04:55We have the number 10 here. 5 multiplied by which number is 10? That is 2. Answer is 2 here. Good to go.
05:05Alright. Now 60 is divided in two parts. 30 multiplied by this number should give you 60.
05:1430 multiplied by which number is 60? That is 2. Answer is 2 here. Likewise, 10 multiplied by 3. It's 30.
05:25Now split 10 in two parts. Which two numbers multiplied will give me 10?
05:30It would be 1 into 10 also, but we are taking the prime numbers. So that is 2 multiplied by 5.
05:38Your answer is now 2, 5. The product is 10. I hope this much is easy enough. Let's move to another question.
05:46Now question number 2 is easy enough that I have just repeated.
05:50Which factors are not included in the prime factorization of a composite number?
05:56For example, I have a composite number 4. Can I write it like, if I repeatedly divide 2 into 2 is 4, 2 into 1 is 2.
06:06But 1 is also the factor. But do we include this 1 here? I do not include 1 ever in the prime factorization as it's not a prime number.
06:17So answer here is 1. Which factors are not included? We never include 1. Reason being, it's not a prime number.
06:28Let's start question number 3. It says, write the greatest 4-digit number and express it in the terms of prime factors.
06:36So it is just like, you know, I mean directly saying do the prime factorization of this number.
06:42The knowledge check is there if you know which is the greatest 4-digit number.
06:47We know this already. Greatest 4-digit number is 9, 9, 9 and 9.
06:53Okay. We have to prime factorize this number as required in the question.
06:58It's your choice if you do through the repeated division of factor tree.
07:02I suggest, unless and until it is not written factor tree, avoid doing that. It's an easy method doing the repeated division.
07:11Okay. Since it's an odd number, we cannot divide it by 2. I'm starting dividing it by 3.
07:189 divided by 3, every time we'll get 3, 3. Now all these numbers are once again divisible by 3. We get 1 everywhere.
07:28Now this number, divisibility rules are going to help you a lot here.
07:34I have made a separate video on divisibility rule. Everyone should go and watch that video.
07:39You'll get a command if this number is divisible by which number.
07:43Now have a look here. 1, 1, 1, 1. It is divisible by 11.
07:49Do not make an error. Generally students do. They give the answer 11. 11 into 1, 11, 1, 11. Have a look.
07:5711 into 1 is 11. Difference is 0 and this 1 will go down.
08:04After taking the difference and bringing the digit down, I have to divide this number by 11.
08:10But it is not divisible. I need one more digit. Whenever you need to borrow one more digit, you need to put a 0 here.
08:19Now easily you can borrow another digit. Right? So 11 into 1 is 11 and it's done.
08:28So we have divided by 11. My number is 101. 101 is itself a prime number. Into 1 and it's done.
08:37How do I know it comes through practice only? I hope everyone is aware between 1 and 100, there are only 25 prime numbers.
08:46So please go and learn those prime numbers. 101 is again a prime number.
08:51So answer of question number 3, how will we write answer of 3? 9, 9, 9, 9.
08:58That is equal to, now write the product of these factors. 3, 3, 11 and 101.
09:07So we have done the prime factorization of the greatest 4 digit number and have a look here.
09:13All these factors are prime factors. Easy enough?
09:18Let's move to question number 4. Exactly a similar question. Now we have to take the smallest 5 digit number.
09:26How do we pick the smallest 5 digit number? Greatest is easy. Every digit will be 8.
09:33When you are taking the smallest number, starting digit has to be 1, rest digit 0.
09:41I have to make a 5 digit number. First digit is 1, rest 1, 2, 3, 4. All are 0.
09:48This is the smallest 5 digit number. Question number 4 we are doing.
09:53Again it has to be written as the product of its prime factorization. So prime factorization has to be done.
10:00It's an even number. So we can start dividing by 2.
10:05I request stop the video, do it yourself and then recheck the question.
10:102 into 5 is 10. All 3 zeros. Now 5000 divided by 2 is 2500. 2500 divided by 2 is 1250.
10:23Again divided by 2 is 625. Now this number is divisible by 5.
10:315 into 1 is 5. 2 into 10 is 5. Again divided by 5 is 25. 25 divided by 5 is 5. Again divided by 5 is 1.
10:44We are not going to take this 1 as it's not a prime number.
10:48How many times 2 is repeated? 4 times. And 5 is also repeated 4 times.
10:55So how will you write your answer? You have to write it in this format.
11:0010000, the smallest 5 digit number is equal to 1, 2, 3, 4.
11:07We have taken 2 4 times and there should be a multiplication sign in between.
11:12Another 5 is repeated 4 times. 1, 2, 3 and 4. So that's your answer.
11:20It is the prime factorization of smallest 5 digit number.
11:25Move to question number 5. It says write all the prime factors of 1729.
11:33Again a prime factorization based question. And you have to arrange the factors in the ascending order.
11:40I request read the question carefully. You'll get to know what all things are being asked.
11:46Step number 1, prime factorization. All the numbers should be prime of course.
11:51Arrange them in the ascending order. Then see if there is any relationship between two consecutive prime numbers.
11:59Consecutive means which come one after the other.
12:03So let's start doing the prime factorization of this number 1729.
12:10Question number 5. We are knowing 1, 7, 2 and 9.
12:16A clue is already there in the question that only the prime numbers will be used to divide it.
12:21It's not an even number. So not divisible by 2.
12:25It's not divisible by 3 either because some of the digits have a look.
12:309 plus 1, 10, 17 and 2, 19. Not divisible by 3, 5.
12:36Let us see if it is divisible by 7.
12:407 into 2. Let me do it here for you. 1729 divided by 7.
12:477 into 2 is 40. Remainder 3. 2 will go down.
12:537 into 4 is 28. Difference 4, this 9 down.
12:58And 7 into 7 is 49. Remainder 0.
13:02That means this number is completely divisible by 7.
13:07I request these questions become easy when we know the tables well.
13:14Tables are going to play a very important role solving here.
13:18Please learn the tables till 20. You can easily solve these questions.
13:22Now this number 247, it's not divisible by 7 once again.
13:28Neither by 11. So next number is 13.
13:32Please have a look. 247 divided by 13.
13:3613 into 1 is 30. Difference 1, 1 and 7.
13:41And 13 into 9 gives you 117.
13:46Now what if you do not know what is the table of 13? You'll get stuck here.
13:50Right? 13 and 19 both have prime number.
13:54So dividing by 13, you get 19 here.
13:58And 19 into 1 is 19. Do not take this one.
14:02So how will you write 1729?
14:051729 is the product of 7, 13 and 19.
14:13Now we have already arranged them in the ascending order.
14:18Question is asking, is there any relationship between any two consecutive factors?
14:23Consecutive means one after the other.
14:26For example, 1, 2, 3. They come one after the other. Consecutive.
14:30Consecutive odd number. 1, 3, 5, 7, so on.
14:35Consecutive even number. 2, 4, 6, 8, so on.
14:40So consecutive means one after the other.
14:437 and 13 are consecutive here in this case.
14:4713 and 19 are consecutive here in this case.
14:52Is there any similarity or any relationship between these factors?
14:57The difference between 13 and 7 is 6. Difference here is 6.
15:02In the similar manner, difference here is 6.
15:05So what you can conclude?
15:07The relationship between two consecutive factors here is this,
15:11that they have a difference of 6.
15:14You can show here, 13 minus 7 is 6 and 19 minus 13 is against.
15:21So that's all about question number 5.
15:24Moving on to question number 6.
15:26The product of three consecutive numbers is always divisible by 6.
15:32You have to verify this statement with the help of some example.
15:36Consecutive, I have just explained you.
15:39Consecutive means one after the other.
15:42But what am I supposed to take the product?
15:45Product means multiplication.
15:47And how many numbers we need to take? 3 numbers.
15:50So it is important to read the question carefully.
15:53Let us see few examples to prove this.
15:57Answer number 6.
15:58Example number 1.
16:00Let me take three consecutive numbers 1, 2, 3.
16:03They are consecutive 3 numbers.
16:06Product means the multiplication.
16:09If I multiply these numbers and divide by 6.
16:133 into 2 is 6.
16:156 into 1 is 6.
16:16Is this 6 divisible by 6?
16:18Yes, it is divisible.
16:20Answer is 1.
16:21Another if I take, let's say 4, 5, 6.
16:26Example number 2.
16:28They are again consecutive 3 consecutive numbers.
16:32I have to take the product and divide by 6.
16:366 up, 6 down.
16:38It is cancelled.
16:39What's your answer?
16:405 into 4.
16:42That's 20.
16:43So we have seen two examples here.
16:46You can take any number you wish to.
16:48Any three consecutive numbers.
16:50And you will find that the product is divisible by 6.
16:54Moving on to question number 7.
16:57The sum of.
16:59Initially in question number 6, we have taken the product.
17:03Product is multiplication.
17:05Question number 7, we are talking about sum.
17:08Sum means addition.
17:10How many numbers do we take then?
17:122.
17:13And the numbers should be consecutive odd numbers.
17:17Please understand each and every term mentioned here.
17:21We have to take how many numbers?
17:242.
17:25They should be consecutive and as well as odd.
17:30And what are we supposed to do with these numbers?
17:33We have to add.
17:34Right?
17:35Again we have to see some example.
17:37Answer number 7.
17:38Example number 1.
17:40First odd number is 1.
17:42Another consecutive is 3.
17:45If I add these numbers, what is the sum?
17:494.
17:50We have to prove that the sum is always divisible by 4.
17:544 is divisible by 4.
17:57Answer is 1.
17:58It is verified here.
17:59Let us take another set of numbers.
18:02If I take first odd number is 3.
18:05Which is the next consecutive odd number?
18:08That is 5.
18:09Question says add the number.
18:11Addition.
18:12The sum is 8.
18:14Now it is also divisible by 4.
18:17Answer is 2.
18:18So we have seen through two examples that a statement holds true.
18:23You can check with another combination of odd numbers.
18:28For example, you can take after 5, we have 5 and 7.
18:33If I add them up, is it giving you 12?
18:36Which is once again divisible by 4.
18:39Answer is 3.
18:40So any two consecutive odd numbers if you add, the sum will always be divisible by 4.
18:47Question number 8 is very easy, oral enough that we have already understood in the beginning of the session only.
18:55In which of the following expression, prime factorization has been done?
19:01Where it is yes.
19:02That means in prime factorization, no composite number should be there.
19:07Have a look.
19:09We have 4 here.
19:10It's a composite number.
19:12Prime factorization.
19:13Nonda.
19:14All the numbers are prime.
19:17And if I see the product also to match.
19:21Triple 8.
19:222 into 2, 4.
19:234 into 2, 8.
19:248 into 7 is 56.
19:26Everything is matching.
19:28This is correct.
19:29C part.
19:302, 5, 7.
19:32All three are prime.
19:345 into 2, 10.
19:3510 into 7, 70.
19:37That is again correct.
19:38Last one we can see 9 is a composite number.
19:42So it is incorrect.
19:44Good to go.
19:45Move to question number 9.
19:48It says 18 is divisible by both 2 and 3.
19:5118 is divisible by 2 and 3 both.
19:55And it is divisible by the product of these numbers.
19:59If I multiply these two, I get 6.
20:02And this 18 is divisible by 6 also.
20:05Question says, if I have the combination of 4 and 6.
20:11If there is a number which is divisible by 4 as well as by 6.
20:17So is it necessary that if I take a product of these numbers.
20:21That is 24.
20:23This number will be divisible by 24 also.
20:26We need to check this.
20:28Every time.
20:29If I am saying yes.
20:30Every time it has to be true.
20:32Let me take an example.
20:35If I take a number 36.
20:38Can it be divided by 4?
20:404 into 9, 36.
20:42Yes.
20:4336 comes in the table of 6.
20:45Divisible by 6.
20:47But if I take the product.
20:49Product is 24.
20:51But 36 is not completely divisible by 24.
20:54So that means it is wrong.
20:56And we have shown the example also.
20:58But why it is true in this case?
21:01It is true in this case because.
21:03When the two numbers are co-prime.
21:07They do not have any common factor between them.
21:10Other than 1.
21:11Only then it is true.
21:13Otherwise not.
21:154 and 6 are not co-prime.
21:17They have 1 and 2.
21:19Both as a common factor.
21:21Okay.
21:22So whenever a combination of co-prime numbers are given.
21:26The product of the number will also divide the given number.
21:30Otherwise not.
21:32Moving on to the last question of this exercise now.
21:36We have to make a smallest number.
21:40Using 4 different prime numbers.
21:43So what we are supposed to do.
21:45Start taking the first 4 prime numbers.
21:48Starting from the smallest one.
21:50We know that the smallest prime number is 2.
21:53Next one is 3.
21:55Next one is 5.
21:57And next one is 7.
21:59Are they 4 prime numbers?
22:01The smallest one.
22:03We have to just take the product of these numbers.
22:06And that is your answer.
22:07Because we have to find the number.
22:10For which we will do prime factorization.
22:12So we have to get 4 different prime factors.
22:15And all should be arranged in the ascending order.
22:18We are making the smallest number.
22:20So take the first 4 smallest prime numbers.
22:23Right.
22:24Another clue.
22:25When there is a combination of 2 and 5.
22:28Make your calculation easier.
22:30Multiply these two first.
22:325 into 2 is 10.
22:343 into 7 is 21.
22:37And what is 21 into 10?
22:39That is 210.
22:40That's your answer.
22:42Right.
22:43So I hope you find this session useful and interesting.
22:47Please don't forget to watch another video.
22:50Where we will be doing exercise 3.6.
22:53And if you have not liked and subscribed.
22:56Please press the like button and subscribe the channel.
22:59Thank you so very much for watching and take care.
23:06See you.