 Hello and welcome to the session. Today I'll help you with the following question which says use Euclid's division lemma to show that the square of any positive integer is either of the form 3m or 3m plus 1 for some integer m. First let's see what Euclid's division lemma is. It says given positive integers there exist unique integers q and r satisfying is equal to bq plus r where r is greater than equal to 0 and less than b. This is the key idea for this question. Let's move on to the solution. Let x be any positive integer. So from the Euclid's division lemma we can say that x can take any of the forms 3q or 3q plus 1 or 3q plus 2. When x is equal to 3q we have x square is equal to 3q the whole square equal to 9q square. Then when we have x equal to 3q plus 1 then x square is equal to 3q plus 1 the whole square which is equal to 9q square plus 6q plus 1. Then we have when the value of x is equal to 3q plus 2 then x square is equal to 3q plus 2 the whole square equal to 9q square plus 12q plus 4. Now we consider the square of the positive integer x often form 3m or 3m plus 1 like when x is equal to 3q its square is 9q square. So we say let 9q square be equal to 3m or 3m plus 1. Thus when we take 9q square equal to 3m from here we get m is equal to 9q square upon 3 equal to 3q square or when 9q square is equal to 3m plus 1 from here we have 3m is equal to 9q square minus 1. Thus m is equal to 3q square minus 1 upon 3. In the same way we take x equal to 3q plus 1 its square is 9q square plus 6q plus 1. Now let's take the square of 3q plus 1 of the form 3m or 3m plus 1 that is we have let 3q plus 1 the whole square equal to 9q square plus 6q plus 1 equal to 3m or 3m plus 1. So when 9q square plus 6q plus 1 is equal to 3m from here we get m is equal to 3q square plus 2q plus 1 upon 3. Or we can say here m is equal to 3q plus 1 the whole square upon 3. In the same way if we take 9q square plus 6q plus 1 equal to 3m plus 1 from here we have 3m is equal to 9q square plus 6q thus m is equal to 3q square plus 2q. Next we have taken x equal to 3q plus 2 now x square is equal to 9q square plus 12q plus 4. Here also in the same way we have 3q plus 2 the whole square equal to 9q square plus 12q plus 4 we take this to be equal to 3m or 3m plus 1. So when we have 9q square plus 12q plus 4 equal to 3m from here we get m is equal to 9q square plus 12q plus 4 upon 3 or we can say m is equal to 3q plus 2 the whole square upon 3. Also when 9q square plus 12q plus 4 equal to 3m plus 1 from here we have 3m is equal to 9q square plus 12q plus 3 thus value of m would be equal to 3q square plus 4q plus 1. So from here it is clear that for any positive integer its square is of the form 3m or 3m plus 1 for some positive integer m. Thus square of any positive integer is either of the form 3m or 3m plus 1 for some integer m. So hope you enjoyed the session have a good day.