 Hello friends, so welcome to another question of the day and this question is what is the remainder when 1989 into 1990 into 1991 plus 1992 to the power 3 is divided by 7 So this question can be categorized as a let's say question of number theory or especially or specifically it is Regarding divisibility in number theory, right? So either you can use modular arithmetic for this or You don't actually require any high-end stuff to solve this problem. So How do we approach? So this is how we have solved it. So 1989 if you notice can be expressed as 7 times 284 plus 1 this is important thing why we have picked up 7 because we will express any number in terms of multiples of 7 and plus some remainder. So basically if you see this is nothing but Euclidean's Euclidean's divisible lemma where a can be expressed as bq plus c where Sorry, not bq plus c bq plus r where r is the remainder and 0 less than equal to r less than b This is what we are going to use. So hence here a is 1989 and b is 7. So if you see 1989 is 7 times 284 plus 1 Similarly, 1990 is equal to 7 into 284 plus 2 and 1991 is equal to 7 into 284 plus 3, right? Now we're saying let's let us say that 284 bk Why I'm assuming it to be k because otherwise it will be too cumbersome to write this number again and again So let it be k. Now, hence 1989 is 7k plus 1 1990 can be written as 7k plus 2 91 can be written as 7k plus 3 and 1992 7k plus 4. Now, let us say the given expression in the arithmetic You know Calculation which is there. So x is equal to let us say this is x x equals to 1989 Plus 1990. I'm sorry. I have written it a little wrongly. So it is not plus its multiplication both here So let us say it is 1989 into 1990 plus into 1991 1992 to the path 3 So expressing this in terms of whatever we just did 7k plus 1 7k plus 2 7k plus 3 and 7k plus 4 whole cube Now you can do the you know simplification. So I have multiplied these two terms first of all So I'm multiplying these two. So opening it you will get this particular term Then 7k plus 3 is as it is and then you use the identity a plus b whole cube to get this Correct. Now, what do we do? So I have just now the next step in this multiplication will be you take this term and you take this term and Multiply with 7k, isn't it? So that term is here Okay, and then 3 multiplied by the entire term again and I have opened up So I have taken this 3 which 3 this 3 and multiply this with the entire this term So you will get this 3 into 7k square plus 9 into 7k plus 6 3 into say 2 is 6 So this is the term and here you have I have written all the 7k terms if you see Together and then the last term 4 to the power cube 3 is 64 Which can be written as 63 plus 1. Why am I writing 63 because I can take 7 come from it You'll see a little while later now Here the first term again in the in the first part of this this term if you see and the entire thing I have taken the 7k thing together. So 7k the first term then all the 7k's are together and then I have left this 6 to be added later on with the 1 which was Which will which is here? Okay, so hence if you express all of them, so you'll see all the terms contain 7. Can you see this 7 here? 7 here 7 here here here here Right and the last two terms are 63 plus 7. Yeah, that is 70 again So if I take 7 common and whatever is the rest I can call it as m then this entire big Expression can be expressed as 7m plus 70. Is it it 70 is coming from this part 70 and 7 times m is coming from the other part, which is remaining over there So then I can write it as 7 times m plus 10 now clearly M is an integer. So m plus 10 is also an integer So hence I can express x as 7 times. So hence x is equal to 7 into some integer An integer. Let's say an integer. Isn't it? That means what this means 7 divides x and Expression is like this. So this whenever you see a vertical bar separating two Expression, you can say you can assume that the left-hand side divides the right-hand side here, right? So 7 divides x. So hence if 7 divides x the remainder must be 0 right? So this is the answer. So when 7 divides entire big expression this one x here remainder is 0