 Hi and welcome to the session. Today we will learn about time factorization. Let us consider a number say 18. Now 18 can be written as 2 into 9. Now 9 further can be written as 3 into 3. So we will get 2 into 9 equal to 2 into 3 into 3. Now we cannot factorize any of these factors further. And 18 can also be written as 6 into 3 and we can factorize 6. So we will get 2 into 3 into 3. Now in both the above factorizations we ultimately arrive at only one factorization that is 2 into 3 into 3. Now here all the factors are prime factors. So this type of factorization of a number is called prime factorization. Thus for prime factorization factorize a number into prime factors. Now there is one more way to find out the prime factorization of a number that is factor 3. Consider the number 60. Now 60 can be written as 6 into 10. So there are 2 factors of 60, 6 and 10. Now 6 can be written as 2 into 3. So there are 2 factors of 6, 2 and 3 and 10 is equal to 2 into 5. So there are 2 factors of 10, 2 and 5. Now we cannot factorize 2 and 5 further. We have prime factorization 60 is into 3 into 2 into... Let's see one more way to find out the prime factorization of a number. Let's take the number 60 again. Now 60 is divisible by the prime number 2. So let's divide it by 2 and we get 30. Now 30 is divisible by 2 again. So we will divide it and get 15. Now 15 is divisible by the prime number 3. So let's divide it by 3 and we will get 5. Now 5 is not divisible by 3 but 5 is divisible by the prime number 5. So we divide it by 5 and get 1. So from here we get prime factorization of 60 is 2 into 3 into 5. Now our next topic is highest common factor or HCF. The highest common factor or HCF of 2 more numbers is the highest, greatest of their common factors. It is also known as greatest common divisor or GCD. Let's take an example for this. Let's find out the HCF of 3 numbers 18, 54 and 81. So first of all let's find out the prime factorization of 18, 54 and 81. Now here we have the prime factorization of 18, 54 and 81. So 18 is equal to 2 into 3 into 3, 54 is 2 into 3 into 3 into 3 and the prime factorization of 81 is 3 into 3 into 3 into 3. Now HCF is the highest or greatest of the common factors. So let's find out the HCF of these 3 numbers. Here the factor 3 is common in all these 3 numbers. Again the factor 3 is common. Now as we can notice that there is no other factor which is common to all these 3 numbers. So the HCF of these 3 numbers is 3 into 3 which is equal to 9. Now our next topic is lowest common multiple or LCM. The lowest common multiple or LCM of 2 or more given numbers is the lowest or smallest or least of their common multiples. Let's take an example for this. Let's find out the LCM of 20 and 50. Now here the prime factorization of 20 is 2 into 2 into 5 and the prime factorization of 50 is 2 into 5 into 5. Now in these prime factorizations the maximum number of times the prime factor 2 occurs is 2 and this happens for 20. Similarly the maximum number of times the factor 5 occurs is 2 and this happens in the case of 50. Now the LCM of 2 numbers is the product of the prime factors counted the maximum number of times they occur in any of the numbers. So here LCM is equal to 2 into 2 into 5 into 5 which is equal to 100. Now there is one more way to find out LCM. So let's find out the LCM of 20 and 50 again. We will write the numbers 20 and 50 in a row like this. Now we will divide these numbers by the least prime number which divides any of the given numbers. Now here 2 divides both these numbers so we will divide them by 2. Now dividing 20 by 2 gives us 10 and dividing 50 by 2 gives us 25. Now here 10 is the multiple of 2 so we will divide these numbers by 2 again. So 10 divided by 2 is 5 and now 25 is not divisible by 2. So 25 will be written as such in the next row. Now next prime number is 3 but 5 or 25 are not divisible by 3. So we will divide them by the next prime number which is 5. Here 5 and 25 both are the multiples of 5. So we will divide them by 5 and we will get 1 and 5. Now again 5 is the multiple of 5 so we will divide them by 5. Now here 1 will be written as such in the next row and here we will get 1. So here the LCM of 20 and 50 will be given by 2 into 2 into 5 into 5 which is equal to 100. Thus in this session we have learnt prime factorization of a number and how to find the LCM and LCM of given numbers. With this we finished this session. Hope you must have understood all the concepts. Goodbye, take care and have a nice day.