 Welcome back. In the last lecture we ended with the divisibility relation. So, let us quickly go through that. Let us just repeat. If you have two natural numbers A and B and if you are able to write A as B into C for yet another natural number, then we say that B divides A. And we will write it in the short form as B vertical bar A. However, while reading that short form we will still read it as B divides A. And in the last lecture we actually saw a quick proof that 1 divides A for every natural number as well as A divides A for every natural number. So, every natural number comes equipped with two God given divisors. There is one and there is the natural number itself. So, let us just go to the next slide where we see that like the subtraction, divisibility also gives you an order. You may wonder that I have been talking about order relations. So, is there a definition of the order relation? Indeed, there is a definition of in fact, not just of an order relation, there is a definition of a relation. What do you mean by two elements in a set being related? Once you see that definition, then the order relation should come with some more properties. It should be a relation with some more properties. So, I will just invite you to go and read somewhere maybe in some basic book or you may want to search on the internet or maybe Wikipedia will help you in that. But those definitions are not important for us at the moment. What we are going to do is concentrate on this very specific order relation, the relation of divisibility. And as we observed in the last lecture also, we do not have a trichotomy here. You can have two natural numbers say 2 and 7. Now, 2 does not divide 7, 7 does not divide 2 and of course, 2 and 7 are not the same. So, given any two natural numbers, we cannot say that a equal to b or a divides b or b divides a. However, when the situation is not normal, that is when things get interesting. So, this is what will give us our basic definition, the most fundamental definition that of a prime integer and then primes come equipped with many fantastic properties as we will see later. So, before going to do that, let us again warm up ourselves with some very basic properties of divisibility. I am going to list them on the next page and just like the last lecture, we will try to prove them one by one. These are the three properties. What does the first property say? It says that if you have a dividing b and b dividing c, then a divides c. This is encoded by saying that the order relation is transitive. If a divides b and b divides some another natural number c, then a should divide c. What does the next property say? It says that if a divides b and I multiply both sides by a natural number c, then the division relation is preserved. The multiplication will preserve the division relation. You may ask, what will happen if I add a number? If a divides b and you add an integer, is it true that a plus c divides b plus c? If I have not written that property, maybe there is a possibility that this property does not hold. Why do not you think about this for a while? After this lecture is complete, you may think about this, does it hold that a divides b and I add any integer c, any natural number c to a and b and yet we have that a plus c divides b plus c. Think about this. What is the third property? Third property tries to combine the divisibility relation with the earlier relation. This is a thing which is omnipresent in mathematics. You will see that everywhere whenever we are developing a theory, we will introduce some concepts and each time we introduce a new concept, we will try to relate it with the earlier concepts. That is what makes mathematics more interesting. You will not see us introducing one concept after another and not seeing the properties of what happens with respect to one concept and the other concept. This is very nice about mathematics. So, the third property says that whenever I have a dividing a b, remember we are within the set of natural numbers. Whenever you have a natural number a dividing a natural number b, then it should happen that a is less than or equal to b. So, it tells you when you turn this statement around, it will tell you that if you have a number a which is bigger than some number b, then this bigger number can never divide the smaller number. The statement I have just said is equivalent to the statement we have seen here. So, there are these three properties. We will try to go over them one by one and prove them one by one. So, here comes the very first property. If a divides b and b divides c, then a divides c. Like our last lecture, I am going to give you a minute to think about this proof and then after that I will give my own proof. Well, my meaning the one that is there in most of the books. If you have any other proof, then the one that I am giving please write to me and let me know your proof. So, I have given you your minute and now let me give you my proof. So, here goes we want to prove that the divisibility relation is transitive. Proof we have equal to a d1. It is given that a divides b. So, b has to be of the form a into some natural number d1 and c equal to b into d2 for some d1, d2 in n. This is what we must have. If you have a divides b and b divides c, then we should have that b is a into a natural number, c is b into possibly another natural number we do not care. So, we have b is a d1 and c is b d2. Then in this equation I will take the value of the b that we have got here. This is a d1 into d2 which is same as a into d1, d2 and note that d1, d2 being product of two natural numbers is a natural number itself. So, we have written c as a into a natural number and hence a divides c. So, we have proved that when a divides b and b divides c, then a must divide c. Let us go to the next property if a divides b and c is a natural number then ac divides bc. So, multiplication by natural number respects the divisibility relation. Once again as we have our deal I will give you a minute to think about this proof. Quite likely you will get the proof that I have in mind which is a very good thing that will tell us that we are thinking in the same direction or even more interestingly you may have a different proof and I would like to know that different proof. So, your minute starts now. So, your minute is up and let me now attempt to give you a proof. We are given that a divides b. So, we have b equal to a into d for sum d in n, a divides b. So, b is a into a natural number. Now to this equation we can multiply by c. So, we have bc equal to ad into c which is same as ac into d. So, therefore we have written bc which is a natural number as ac into a third natural number which clearly says that ac divides bc. Whenever a natural number a divides another natural number b and to both sides you be fair to both the natural numbers which are there on either side of the vertical bar. You multiply by a natural number to both these natural numbers then the divisibility relation is respected. We now go to the third property which is a very important property and we will use it quite often. This says that if you have a natural number a dividing a natural number b then with respect to the earlier order that we have defined less than it should be less than or equal to b. This strange symbol that you may see here means that either a is less than b or a is equal to b. Once again I will give you a minute to think about this proof and after your minute is done I will state my proof. Your minute is up. So, let us see the proof. This proof is quite interesting. How would we prove this? So, let us see. So, since divides b we have b equal to ad sum natural number d. So, we have b as a product of a and a natural number d. Now this natural number d that you have got could either be equal to the number 1 or it could be bigger than 1. So, if d is 1 then b is a into 1 which is a. This is one case that b is equal to a. Now if I take d bigger than 1 then b is a plus a plus a plus a. How many times do we get this? This is d times. We have that b is a into d. So, b is a plus a plus a d times. What we want to prove is that if you have a dividing b then a is less than or equal to b. This is what we want to prove and we have already seen that when d. So, b is a into d if d is 1 then a equal to b. If d is bigger than 1. Observe first of all a plus a is bigger than a. If I have a natural number a and I add the natural number to itself I get something which is bigger than that and then inductively I get a I am adding a to both these sides. And we know that the addition respects the bigger than sign or the less than sign. But what is also true is that it preserves the transitivity is preserved. That means if you have a natural number which is bigger than second natural number which is bigger than the third natural number. Then we see that the difference of the first one and the third one is a natural number and therefore we have transitivity. This way d into a I will just write it as d into a which is a plus a plus a d times is going to be bigger than a and remember our b was d into a or a into d. This completes our proof. Once again there were two possibilities either you could write b as a into 1 in which case you get equality or you could write b as a into a d which is bigger than 1. And then we get that b is strictly bigger than a it is not equal to a but it is strictly bigger than a. So, let us just repeat what we have proved until now. We have proved that when a divides b and b divides c then a divides c. This says that the divisibility relation is transitive. Further if you have a dividing b and you multiply both sides by a natural number c then we will have that ac divides bc. And finally the most important property is when a divides b then a has to be less than or equal to b. These three are very important. There might be some extensions of these properties for the set of integers for the set z. In fact, you could go all the way up to the set of real numbers, complex numbers or if you know some advanced mathematics the number of the quaternions what are also known as Hamiltonions all the way up to wherever you can go. But at the moment what we have is the following. If you have two natural numbers then we say that one natural number divides the another natural number. If there is a third natural number such that the first one can be the second one can be written as the first into the third. Now, brace yourself we are coming to the most important definition in the basic theory of numbers which is as follows. This is the definition of a prime. Really the English word prime has very good meaning it is you know something which is in the best of its situations. And prime number is really the best thing that you can have in the number theory. So, what is the prime? We have observed that whenever you take any natural number a there are two God given divisors one divides it and the number a divides it. If your natural number happens to be just the number one then there is only one God given divisor because both these divisors happen to be the same. In any case any number bigger than one is going to have two divisors which are given to us always and we say that a natural number is a prime if there are no further divisors. So, we say that our natural number p is a prime if there are exactly two divisors not one, not three and not further no 4, 5, 6, 7. So, once we have made a definition a natural thing would be to see some properties maybe prove some results but before that we should see examples. Let us go over our numbers we are working with natural numbers. So, we will start from one and go onwards. Let us start from one go onwards and see if we get a prime is one a prime. What would you think? How many divisors are there for one? We saw just now that for one there are no two divisors because for any integer for any natural number a there are two divisors one and a but one unfortunately has the property that both these divisors become the same. So, the number of divisors of one in n is one. One is the only unfortunate number in this way. So, one cannot be a prime one is not a prime. What about the next number? Is two a prime? How many divisors are there for two? One is a divisor of two. Two is a divisor of two. Is three a divisor of two? What did we learn in the last property of divisibility? If three divided two then three would have to be less than or equal to two. But three minus two which is a natural number one tells us that three is bigger than two. Similarly, four, five, six, seven all numbers are bigger than two. So, if you are looking for divisors of two you should go no further. This is something you should take a note of. If you are looking for divisors of a number a you need to go only up to a. You do not have to consider the natural numbers from a plus one onwards. So, it has become a finite problem. To find the divisors of a you have to go no further than a. In fact, you can make this problem simpler. You need to go only up to the square root of a but that will come when we deal with really large numbers. So, is two a prime? We have counted the number of divisors of two. One is a divisor and two is a divisor and there are no further. So, two is indeed a prime. What about three? Is three a prime? One divides three, three divides three. We need to look no further. So, four onwards are the natural numbers that we will ignore. There is another number sitting between one and three which is two. Does two divide three? How do they multiples of two look like? Two into one is two. Two into two is four. Two into three is six. So, the higher natural number you multiplied by to two you get bigger and bigger numbers. So, two into a natural number is not equal to three. Therefore, two does not divide three. So, the divisors of three are one and three nothing further. So, three is also a prime. What about four? Can you think about four being a prime? We will need to look at the divisors of four. One is a divisor, four is a divisor but we have counted the divisors of the multiples of two just now and we saw that two into two is four. So, four has three divisors. One, two and four itself. So, four is not a prime. All right. So, we got one is not a prime, two is a prime, three is a prime, four is not a prime. You may continue this way. Can we list all primes up to 100? Well, the way we have done so far, this should be a work which is possible. So, yes there are 25 of them. Can we list them all? 2, 3, 5, 7, 11. This is the next bunch. This is the next bunch. This is the next and this is the last bunch. So, these are exactly the primes up to 100. You know, you would see a teacher in your class giving you a problem and if you solve it very easily, the teacher will give you a slightly harder problem. I am going to do the same. Can we list all primes up to 1000? The answer is yes. There are 168 of them. The number of primes up to 100 was 25. The number of primes up to 1000, we have multiplied 100 by 10. So, we got 1000. The number 25 did not get multiplied by 10. It got multiplied to somewhere between 6 and 7. So, we got 168. I am not going to write all those primes here. I will give that to you as an interesting homework. Do not send that list to me. Keep it with you. Can we list all primes up to 10000? Yes, we can. I can tell you the number. There are 1229 of them. Once again, we went from 1000 to 10000 by multiplying by 10 and the number 168 did not get multiplied by 10. It got multiplied by a number which is slightly smaller than 8. So, after 10000 you may ask what about 100000? What about 100000? Or can we just go on and on? Can we just go on and on? And the answer is yes. We can. What this means to say is that there are infinitely many primes out there. There are plenty of primes. The set of primes is not finite. With this punchline we will stop here in this lecture and we will state the result of the infinitude of primes in the next lecture and also see a proof. Thank you.