 Hello friends, so in this Session we are going to deal with another question which requires application of Euclid's division lemma now What was Euclid's division lemma? You know that if there is there are two integers a and b If there are two integers a and b then we get a pair of another positive integers Qnr so a and b are positive integers mind you and Qnr are other set of positive integers and then and By Euclid's division lemma, we know that a can be expressed as bq plus r where where a is the dividend b is the divisor q is the quotient and r is the remainder and this condition holds Zero is less than equal to r less than equal to sorry less than b now using we'll be using this application application, sorry application of this lemma here in this question So let's drill down and try to dissect the question. The question says show that the square. So this is important Information square of any positive again positive integer. So square of any positive integer is Of the form of 3m or 3m plus 1 so any any square if you take let's take some examples first So let's say if you have Three and square it so three square is nine. So clearly nine is of the form of 3m where M is three. So again take a random number. Let's say five Squared it you'll get 25 and it is if you see this is three into eight plus one Which is of the form 3m plus one another number. Let's say you take 12 so 12 square is one four four So if you see this itself is a multiple of three 48 Okay, so either it is of the form of 3m or it is of the form of 3m plus one which is Which is you know Clearly seen here, but is this true for all the numbers? That's what the question asked that if you have a positive integer Let's say a then square of Square of a that is a square will always be of the form 3m or 3m plus one. Let's let's now understand how to prove it now We know that any given integer a can be of the form of clearly 3k So if you divide any integer by three you'll get either remainder as zero or a could be or a could be 3k plus one or A could be 3k plus two Correct now How do we know that why because if let's say any integer a can be expressed as three into q plus R where R will be R will be greater than equal to zero, but less than equal to Three three is the divisor here. So what values R can take R can take R can become zero or R can be one Or R can be two so hence using this this is from what Euclid's division lemma If you see so hence any integer can be expressed as 3k plus zero 3k plus one and 3k plus two Let us say case one this this thing. I am saying this is case one So case one when let's say any integer is of the form of 3k plus one. Sorry 3k plus zero This is let's say this is 3k Sorry, this is case two. So This is case two case Two and this one is case three is three. Okay. Now. Let us evaluate All these three class cases and see that in every case The square a square will be of this form. Let us see case by case. So let us take first case one Yeah, so let us take case one. What was case one case one? So case One was when a is equal to 3k. So a square clearly will be 3k whole square Which is 9k square, which can be written as three times 3k square Three times 3k square. This dot is not for decimal. This is for duplication. So three times 3k square So clearly you can say three into M Where where M is equal to 3k squared Okay. Now, let us take so in this case this this work So when a is 3k, definitely a square will be of the form 3m now take case two Case two. So what is case two when a is of the form? 3k plus one. Let us square both sides. You'll get a square and 3k plus one whole squared, which is nothing but if you see 3k whole square plus twice of 3k times one Plus one square, which is from what which is from the identity you all know a plus b whole square is a square plus twice a b plus b square correct now from here If you expand it, you'll get 9k square plus 6k plus One if you see you can take three common from this term. So it is 3k square Plus 2k. So I am taking three common between first two terms plus one So clearly if you if you say let's let us say M is equal to 3k square plus 2k if I assume that this is M then this becomes 3m plus one so This is this satisfies the second case. So either it should be of the 3m form or 3m plus form So this also works. Let us go and see if it works for case three as well So what was case three case case three always it is better to write in These steps in these cases. So let's say a is equal to 3k plus 2 okay, this is the third case square both sides again So a square is nothing but 3k Plus 2 whole square again use identity which one this one use this identity to expand it and you will get what? 3k 3k whole square plus 2 into 3k into 2 Plus 2 square, okay, so simplifying you'll get what 9 9k square plus what? 2 to 4 4 into 3 12k Plus 4 which can be written as 9k square plus 12k plus 3 Plus one isn't it you can write that now from these three terms What do we see we see that there is a common? Factor and that's three. So let's take that common factor out and within parentheses. We can write 3k square plus 4k plus 4k Plus one and then this is extra one Again, so if you consider if you call 3k square plus 4k plus one equals M Then what will you get you will get? What will you get this will give you 3m again plus one What have you assumed we have assumed that m is equal to 3k square plus Sorry 4k 4k plus one Right, so in all the cases in the first case we saw what did we see? We saw that if it is of the 3k form then it is 3m form the square is of 3m when it is of 3k plus one form when a is 3k plus one form a Square is 3m plus one form and when a is 3k plus two form Then a square will be 3m plus one form So in all the three cases we could do that Why did we start and why did we choose B as 3 or the divisor s3 clearly? Why because there is you know the question is saying that that all the Square of terms can be expressed as 3m and 3m plus one so our natural choice of divisor in this case was 3 so that we could get factors of 3 and hence we could prove that yes indeed Square of any integer positive positive integer is either of the form 3m or 3m plus one These are the two possibilities of square of any integer Okay, so This is all for this session. Thanks for watching this do subscribe our channel for more such sessions and We'll be sticking up a few more questions in next videos. Thanks a lot