 Hello and welcome to the session. In this session we discussed the following question which says show that out of M, M plus 2, M plus 4, 1 and only 1 is divisible by 3 where M is any positive integer. So for a positive integer M, we have to find out that out of M, M plus 2 and M plus 4, only 1 is divisible by 3. So before proceeding with the solution, let's recall the Euclid's division lemma. According to this we have given positive integers a and b there exist unique integers q and r satisfying a is equal to bq plus r where this r is greater than equal to 0 and less than b. This is the key idea that we use in this question. Now we move on to the solution. Now when M is divided by 3, then we suppose let q be the quotient, r be the remainder, then we have M is equal to 3q plus r where this r is greater than equal to 0 and less than 3. We have applied the Euclid's division lemma here and so we get M is equal to 3q plus r when we divide M by 3 and this r is greater than equal to 0 and less than 3. So this means M is equal to 3q plus r where r is equal to 0, 1 and 2. Now when we put r equal to 0, we get M equal to 3q and when we put r equal to 1, we get M equal to 3q plus 1 and when we put r equal to 2, we get M equal to 3q plus 2. So for r equal to 0, 1, 2 we get M equal to 3q or M equal to 3q plus 1 or M equal to 3q plus 2. So now let's consider the case 1 in which we have if M is equal to 3q, then this means that M is divisible by 3. So in the first case, we take M equal to 3q and so this means that M is divisible by 3. Now let's take the case 2 and we have M is equal to 3q plus 1, then M plus 2 would be equal to 3q plus 1 plus 2 which is equal to 3q plus 3 that is M plus 2 would be equal to 3 into q plus 1 and this is obviously divisible by 3. So in this case, we get M plus 2 is divisible by 3 because when we take M equal to 3q plus 1, we get M plus 2 as 3 into q plus 1 and this is divisible by 3. So here we get M plus 2 is divisible by 3. Now consider the case 3 where we take M equal to 3q plus 2, then in this case M plus 4 is equal to 3q plus 2 plus 4 which is equal to 3q plus 6 and so this means that M plus 4 is equal to 3 into q plus 2 and this 3 into q plus 2 is also divisible by 3 therefore M plus 4 is divisible by 3. So on taking M equal to 3q plus 2, we get M plus 4 equal to 3 into q plus 2 and this is divisible by 3. So M plus 4 is divisible by 3. So now we get when M is equal to 3q, then M is divisible by 3, when M is equal to 3q plus 1, then M plus 2 is divisible by 3 and when M is equal to 3q plus 2, then M plus 4 is divisible by 3. Hence we say 1 and only 1 out of M plus 2, M plus 4 is divisible by 3. So we have proved this, with this we complete the session, hope you have understood the solution of this question.