 Well hello dear students, this is Praveen Agrawal, your math teacher at Centrum and let's look at the question number 7 of IOQM that is Indian Olympiad Qualifier for Mathematics 2022 question. It's a very interesting question of basic number theory. It needs no advanced concepts other than you know 10 standard. So it says that find the number of ordered pairs A, B such that A and B both belongs to the set 10, 11, 12 till 13 and also satisfying the equation that GCD of A, B plus LCM of A, B is equals to A plus B. So let's start with assuming that the GCD, let the GCD of greatest common teacher of A, B is small g. Then both A and B will be some multiple of this GCD. So let us assume that our A is some x times g and our B is some y times g. Now if A and B are like that where g is the GCD what will be LCM? Yes, you get it right LCM of A comma B should be equals to x into y into g because LCM contains all the you know the least common multiple of A and B means if you find all the multiples of A and all the multiples of B it will be common multiple of them. There are couple of ways of finding it one of the ways is you know that LCM into GCD will be equals to A into B. Now LCM into GCD is g, A is your you know x times g, B is your y times g. So if you cancel this g you get LCM is equals to x into y into g. Now let me substitute each of these into the given equation which is a GCD of A, B is g plus LCM is x, y, g and on the right hand side you have x, g plus y, g. You know that each of these is going to be a natural number so g cannot be zero so we can very well cancel this g and if you cancel this g what you get is 1 plus x, y is equals to x plus y. If you rearrange this equation you can write it as x minus 1 times y minus 1 is equals to 0. So one of the two things will happen either x equals to 1 or y equals to 1. What does it mean if x equals to 1 then what will happen to A? A will directly be the GCD and B will be y times g. So we can say that B will be a multiple of A or if y is equals to 1 then you can say that A is a multiple of B. So one of the two things will happen either in the ordered pair A, B. A is a multiple of B or vice versa. Alright so some obvious pairs are all the pairs A, A. All the obvious pairs A, A such that A belongs to the set 10, 11 and so on till 30. All of them in all of them they will satisfy this condition and they are both x and y will be equals to 1. A will be a multiple of B as well as B is a multiple of A. So all the pairs like 11 comma 11, 12 comma 12 and so on till 30 comma 30 will be there. How many pairs are these? These are so not just 10, 11 comma 11 it will start from 10 comma So these are 21 such pairs. Next we can take pairs where we take multiples of 10 other than itself. Okay what are the multiple of 10s other than itself? We have 10 comma 20 and we have then 10 comma 30 also. Next let us take now because A comma B here B is a multiple of A you will take the pairs which are the converse of it 20 comma 10 and 30 comma 10 as well. Here A is a multiple of B. Here B is a multiple of A. Here A is a multiple of B. So we will take both of these. So let us just count these ones and we will double all of these okay where A is not equals to B. Next we will have 11 comma 22 and it is converse. Next we will have 12 comma 24. Next 13 comma 26. Next 14 comma 28 and next 15 comma 30. So these are 1, 2, 3, 4, 5, 6, 7, 7 pairs and exactly 7 converse all right. For example for corresponding to 15 comma 30 the converse will be 30 comma 15 so 14 these pairs. Total number of pairs will be 21 plus 14 that is 35. And as you know that in all the 30 questions of IOQM the answer is up to 2 digit integer and 35 is the correct answer. All the best friends keep learning and because the match Olympias of this year is going to take place in September 1st week. Only two months are remaining so keep preparing for that okay. Stay tuned to this channel. All the best.