PLAYING WITH NUMBERS| CHAPTER 3| Ex 3.4 | NCERT | INSIGHTFUL MATHS
PLAYING WITH NUMBERS| CHAPTER 3| Ex 3.4 | NCERT | INSIGHTFUL MATHS
Transcript
00:00Hi everyone. Welcome to Insightful Maths. In today's session, we are going to discuss
00:06the topic Common Factors and Common Multiples using the exercise 3.4 of Grade 6. So, let's
00:13start the session.
00:17Common factors if we talk about, factor is what? Factor is that number which divides
00:24the number completely. For example, if I have a number 12. If I have a number 12, then all
00:33the numbers which divide 12 completely are the factors of 12. So, which is the smallest
00:39factor, the smallest number which divides 12? The smallest factor of in fact all the
00:45numbers is 1. It's an even number, divisible by 2. It is divisible by 3, divisible by 4.
00:54It does not come in the table of 5, nor divisible by 5. It is divisible by 6, not by 7, 8, 9,
01:0310, 11. But 12 comes in the table of 12. So, what is the highest factor of 12? It is 12.
01:09So, the smallest factor of any number is 1 and the greatest factor of the number is number
01:15itself. So, if I talk about the first question, we are looking for the common factors. That
01:21means the numbers which divide 20 as well as 28, both the numbers. So, have a look here.
01:29Let me first write all the factors of 28 and 20. Easiest method, if you are very good at
01:36tables, you can write that directly. Otherwise, for the bigger numbers, there is a scope of
01:41missing the factors. So, what we can do? Write it in the product form. What does it mean?
01:4728, can it be written as 1 into 28? That is 28. Now, 2 multiplied by 14, that is also 28.
01:5828 does not come in the table of 3. Does it come in the table of 4? 4 into 7 is 28. That's correct.
02:08It does not come in the table of 5. It does not come in the table of 6. In the table of 7, 7 into 4 is 28. Now, stop here. Why?
02:17Because all these numbers, 1, 28, 2, 14, 4 and 7, all these numbers, the product of which is 28,
02:27these all are the factors of 28. When do we understand we have to stop? When the numbers start repeating themselves.
02:344 multiplied by 7, I have already got. Now, the numbers started repeating. So, this is a clue that we are done with all the factors and we need not to go further.
02:46So, which are the factors of 28? Let me write it here. 1, 2, 4, 1 to 4. I am writing it in the ascending order, 7, 14 and 28.
02:59In the similar manner, let me find the factors of 20. 1 into 20, 2 into 10 does not come in the table of 3. 4 into 5 is 20 and after 4, 5 into 4 is 20.
03:14Are the factors started repeating? Yes. So, this is a clue that I need not to go further and I have to stop here.
03:22So, I am just not considering this one. It is a repetition. So, what all are the factors of 20? Ascending order, 1, 2, 4, 5, 10 and 20.
03:35We have got all the factors. Question says we have to find the common factors. Common means the numbers which are factors of both.
03:43Have a look. 1 and 1 is common, 2 and 2 is common, 4 and 4 is common. Is there any other number which is common? Nothing.
03:52So, answer is common factors of 20 format 28 are, which are the common factors? 1, 2 and 4. That's what I will say.
04:06Interesting. Let us do one more part. Then these two parts you will be doing yourself as a practice questions.
04:14Okay. The more you practice, the more you will gain the confidence. It is about the understanding of the topic then.
04:22Okay. After that, it is just a practice. So, we have two numbers. B part of question number 1 I am doing.
04:3016 and 25. 1 into 15 is 15. It doesn't come in the table of 2. 3 into 5 is 15. Doesn't come in the table of 4. It comes in the table of 5. 5 into 3.
04:43But numbers started repeating. So, please do not go ahead. We have got how many factors for 15? 1, 3, 5 and 15. Let me write here.
04:551, 3, 5, 16. Now, we have to write for 25. 1 into 25. It is not coming in the table of 2, 3, 4. It comes in the table of 5. 5 into 5 is 25. Okay.
05:11Now, please observe 5 and 5. It's common here. It has started repeating at this level only. Okay. The numbers started repeating at the same level.
05:24It is a clue. I need not to go further. You can check it yourself. 6, 7, 8, 9, 10, 11. Till 24, 25 does not come in any table. It is now divisible by 25 only.
05:37And that number we are already having. So, we just have 3 factors of 25. 1, 5 and 25. Why I am not considering this? Because 5 I have already taken.
05:48There is no use of repeating the numbers. Okay. So, factors of 25 are 1, 5 and 25. Which numbers are common in both? Have a look. 1 and 1 is common. 5 and 5 is common.
06:03Nothing else. So, you have to write common factors of 15 and 25 are. Which are the common factors? 1 and 5. That's your answer. Easy enough. Good to go. Let's go further.
06:20Q2. Exactly similar. Only 3 numbers are given. Okay. Let me pick the B part. Why B? Because the factors of 15 and 25 we are already having on our screen. We just need to find the factors of 5. Okay.
06:37So, Q2. B part now. And make it Q3. 15 and 5 we already have. We have to find the factors of 5. 5 is a prime number. And all the prime numbers have just 2 factors. That means 1 and the number itself.
06:59We have just 2 factors of 5. It's being a prime number. 1 and 5. 1 and the number itself. So, after this I will write 5. 5 the factors. 1 and 5. Which numbers are common in all 3? Have a look. 1 is there in all 3. And 5 is again there in all 3. So, answer remains the same.
07:23Common factors of 5, 15 and 25 are 1 and 5. Easy enough. It is just about finding the factors first and then seeing which number is common in all. Understood? Can we go ahead? Okay.
07:42Okay. Now coming on to the next topic which is the common multiples. Factors we understand. Factor is a number which divides the given number. Now multiple is a number which we get after multiplying the given number with the set of natural numbers.
08:01That means or in the simplest manner you can remember all the numbers which come in the table of a particular number. But table doesn't stop at 10 times. It is 11 times, 12 times, 30 times. It is infinite. So, factors are limited, countable or finite. Multiples are infinite. Okay.
08:23So, question says we have to find the first 3 common multiples. Multiples are infinite. We cannot give all multiples. That is why the question will always specify how many multiples we need to tell. So, let us write the multiples of 6 and 8.
08:53Let us write the numbers in 6 table. 6, 12, 18, 24, 30, 36, 42, 48 and so on. Let's write the table of 8 now. The multiples 8, 16, 24, 32, 40, 48 and so on.
09:21Now, you know what is that trick? Whenever you are trying to find a common multiple of 2 numbers, always start with the multiples of bigger number. After writing multiples, we are looking for the common numbers which come in the table of both. Always start searching the multiples of bigger number in the smaller table.
09:45What does it mean? Let me start finding the multiples of bigger number. 8 comes in the table of 6. It is not there. Put a cross. Do we get 16 in the table of 6? It's no. Put a cross. Do we get 24 in the table of 6? Have a look. We have 24. Do we get 32? No. Do we get 40? No. Do we get 48? Have a look. It is here.
10:12Now, once you get even the first multiple, it is very easy to give the next 4 or infinite multiples. How is that? Please observe the pattern. Till now, we have got common multiples 24 and 48. Is there any relation between these 2 numbers?
10:3348 is 2 times of 24. So, this is the first multiple of 24 which is 24. This is the second multiple. 24 into 2 is 48. So, it should be the third number. It is the third multiple of 24. No oral calculation. Always make a rough column. Do the calculation here and then write your answer.
10:594 into 3 is 12. 3 into 2 is 6 plus 1 is 7. Even if you continue the series, you will get the same answer. But, there is no logic writing it too far when we know the trick. So, the third number is 72 and that's your answer.
11:16What we did? We have written few multiples of 6. We have written few multiples of 8. We started searching the multiples of 8 in the table of 6. First number we got is 24. Second is 48. Otherwise, second doesn't match. I know. If the first multiple I got, now all the numbers in the table of this number will be the common multiples.
11:44Understood? Let me do this part also for you.
11:49Once again, we have to find the common multiples of 12 and 18. What we need to do? Let us start writing the multiples of 12.
12:00Multiples of 12. We are writing 12 into 1 is 12. 24, 36, 48, 60, 78 and so on. I request, please learn the tables at least till 20. It is going to help you a lot in your calculations.
12:22Now, multiples of 18. I know many of you might not be knowing the table of 18. I will go slow. Please calculate it yourself and then write the answer.
12:3418 into 1 is 18. 18 into 2 is? What is the double of 18? That is 36. What is 3 times of 18? Please calculate. 18 into 3. How much is that? That is equal to 54.
12:54What is 4 times of 18? Yes. What is 4 times of 18? I will give you a trick. Even if you don't remember, we are calculating 4 times of 18. Split 18 into 2 parts. It is 10 and 8. Multiply both by 4 and add.
13:1210 into 4 is 40. 8 into 4 is 32. What is the sum? Is it 72? That's your answer. What is 18 into 5? Again, 18 into 5. 18 is split into 10 and 8. Multiply both by 5 and add. It is 50. It is 40. What is the sum? Is it 90? That's your answer.
13:32Start finding the multiples of bigger number in smaller numbers table. Bigger number is 18. Does 18 comes in 12 tables? No. Does 36 comes? Yes, I can see 36 here.
13:50Now, all the numbers in the table of 36 are the forman multiples. How many do we need to give? We need to give 3 forman multiples. So, you have to write forman multiples are. First one is 36. What is the double of 36? I should not say double. Let me ask you to calculate in this manner only.
14:18Write the table of 36 3 times. 3 times means still 3rd position. 36 into 1 is 36. 36 into 2 is what? 6 into 2 is 12. 3 into 2 is 6 plus 1 is 7. Is it 72? Write 72 here. Likewise, 36 into 3. It is 18. 3 into 3 is 9 plus 1 is 10.
14:44So, this is 108. Now, stop here. We were asked to find only 3 multiples and that's your answer. What did we do? Write the multiples of both the numbers. Start finding the multiples of bigger number in the smaller one. Once you get the first forman multiple, just continue the table of that number. Good to go?
15:06Now, write all the numbers less than 100 which are forman multiples of 3 and 4. We have to write the forman multiples of 3 and 4. But what is the condition given here? The answer or the number should be less than 100.
15:29So, question number 4, multiples of 3, multiples of 4. Let's start writing. Multiples of 3 are 3, 6, 9, 12, 15, 18 and so on. Write the multiples of 4.
15:59Start finding the multiples of smaller number in the bigger one. 4 does not come in the table of 3. 8 does not come but 12 comes. Now, what we are going to do? Now, the table of 12.
16:24All the numbers in the table of 12 are the forman multiples of 3 and 4. Let us continue the table of 12 till the time we reach a number which is lesser than 100. It should not go above 100.
16:55But 108 will be exceeding 100. So, we have to stop here. So, this is your answer. All these numbers are the forman multiples of 3 and 4.
17:08Moving on to the next question, question number 5. You have to find which all numbers are co-prime. Now, what are co-prime numbers? Have a look at the definition.
17:20The numbers which have their highest O1 factor as 1. Those are known as the highest O1 factor. Those are known as co-prime. Now, what does it mean? Have a look at question number 5 A part.
17:4418 and 35. What are the factors of ET? Or rather than writing all the factors, we know how to calculate HCF. I have made a separate video already how to find LCM and HCF. If you still do not know, I request please go and watch that video. You will gain the clarity how to find the highest forman factor.
18:09I am using the division method, the easiest method for finding the highest forman factor. If highest forman factor is 1, these numbers are co-prime, otherwise not. So, question number 5 A part. 18 and 35. Two numbers are there. I am finding the highest forman factor.
18:3018 x 2 is 36, which is more than 35. I wrote in first place. It is 18. Take the difference of 35 and 18. How much is that? We will take a carryover. This is 2 and this is 15. 15-8 is 7. 2-1 is 1.
18:51Now, what we are going to do? We have to keep on dividing till the time the remainder becomes 0 and we need not to take the decimal here. Now, this remainder will become the new divisor. Earlier divisor was 80. Now, this remainder becomes your divisor and it will divide the previous divisor.
19:13So, 17 will be dividing 80. 17 x 1 is 70 and the remainder is still 1. Now, this remainder is the new divisor. Now, 1 x 17 is 17 and my remainder is 0.
19:36In the last, with which number I have divided to get the remainder 0? It is 1. This is your highest forman factor. Highest forman factor of these two numbers is 1. Answer is yes, they are co-prime.
19:51I will do another one for 15 and 37. Please practice this method. This is really going to help you a lot. Ok, I will do the B part. 15 and 37. B part now. 15 and dividing 37. 15 x 2 is 30. So, twice it is 30.
20:17What is the difference? It is 7. Remainder is not 0. Now, this remainder will divide the previous divisor 15. 7 x 2 is 14. Remainder is 1. It is still not 0. It will become your divisor and it will divide the previous divisor 7.
20:381 x 7 is 7. Wow! And the answer is 0. What is the highest forman factor now? The last number which is giving me the remainder is 0. It is 1. So, highest forman factor of 15 and 37 is 1. So, are they co-prime? Answer is yes, they are co-prime as the highest forman factor is 1.
21:03In the previous one also, we have seen that highest forman factor was 1. So, these two are again co-prime. I request, do other parts. Find their highest forman factor. If the answer is coming 1, these two numbers are co-prime. Otherwise, not.
21:22Coming to question number 6.
21:24A number is divisible by both 5 and 12. By which other number will that be always divisible?
21:34This is something you need to understand if there is a number. It is divided by both 5 and 12. So, this number will always be divisible by the product, the multiplication of that number as well.
21:52If it is divided by both 5 and 12, then it will always be divided by the product of these two numbers.
22:23So, you just need to remember it.
22:25If these two numbers are co-prime and a number is divisible by 5 and 12 both, this number will be divisible by the product of these two numbers. So, answer here is 60.
22:41The reverse is also true. If a number is divisible by 12, then by what other numbers will it be divisible?
22:52Please understand, a number. If a number is divisible, let me just take an example.
22:58I have a number 24. If I say it is divisible by 12, I need to find by which all numbers it will be divisible. It will be divisible by all the factors of this number 12.
23:16Factor means the numbers which are divisible, which divides 12 completely.
23:22Now, what are the factors of 12? 1, 2, 3, 4, 6 and 12. Let us recheck.
23:33Is 24 divisible by 1? Yes. Is 24 divisible by 2? It is an even number. Yes.
23:3924 comes in the table of 3, 4, 6 and 12. It is divided by all. Hence, we have verified.
23:47If a number is divisible by one particular number, I have a number A. It is divisible by B.
23:56So, how will it be divisible? By all the factors of B.
24:02Understood? One example, if I say I have a number A. It is divisible by 15.
24:11So, it will be divisible by which all other numbers? All the factors of 15.
24:18And we know that which are the factors of 15? It is 1, 3, 5 and 15.
24:26If a number A is divisible by 15, this number A will also be divisible by the factors of 15.
24:36I hope you have enjoyed the topic.
24:39If yes, please like and subscribe my channel for getting the information and the tutorials for the upcoming exercises also.
24:493.5 continued in the next chapter.
24:52Thank you so much for watching.
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