 Hi and welcome to the session. Let us discuss the following question. Question says, if 3 1 z 5 is a multiple of 3, where z is a digit, what might be the values of z? Now before moving on to the solution, first of all let us understand that a number n is divisible by 3 if the sum of its digits is divisible by 3. We will use this property as our key idea to solve the given question. Let us now move on to the solution. Now we are given in the question that 3 1 z 5 is a multiple of 3 or we can say 3 1 z 5 is divisible by 3. Now since 3 1 z 5 is divisible by 3, so sum of the digits 3 1 z and 5 should be divisible by 3. So we can write since 3 1 z 5 is divisible by 3 sum of its digits that is 3 plus 1 plus z plus 5 should be divisible by 3. Now 3 plus 1 is 4 and 4 plus 5 is 9. So we can write that is 9 plus z should be divisible by 3. This is possible only when 9 plus z is equal to any multiple of 3. So we can write this is possible when 9 plus z is equal to 3 or 6 or 9 or 12 or 15 or 18 and other such multiples of 3. Now clearly you can see adding something in 9 does not gives a number less than 9. So 9 plus z can't be equal to 3 or 6. Now let us take the value of 9 plus z as 9. When 9 plus z is equal to 9 then on subtracting 9 from both the sides of this equation we get value of z as 0. Now here z represents a digit 0. Now let us take 9 plus z equal to 12. Now when 9 plus z is equal to 12 then on subtracting 9 from both the sides of this equation we get value of z as 12 minus 9 that is value of z is equal to 3. Now this is again a digit lying between 0 and 9. We know that z is representing a digit so its value must lie between 0 and 9. Now let us take 9 plus z equal to 15. Now when 9 plus z is equal to 15 then on subtracting 9 from both the sides of this equation we get z as 15 minus 9. Which is further equal to 6. So we get value of z here as 6 this is again a digit that is lying between 0 and 9. Now let us take 9 plus z equal to 18. Now when 9 plus z is equal to 18 then on subtracting 9 from both the sides of this equation we get the value of z as 18 minus 9 that is 9 only. So here we get value of z as 9. Now for all other higher multiples of 3 we will get value of z as greater than 9. And we know that since z represents the digit it cannot have value greater than 9. Now we can write since z is a digit therefore z is equal to 0 or 3 or 6 or 9. Thus z can have any of the four different values. How far z is equal to 0? Number is 3105 and 3105 divided by 3 gives 1035 or you can say 3 multiplied by 1035 is equal to 3105. So clearly we can see 3105 is a multiple of 3. Now when z is equal to 3 then the number becomes 3135 and 3135 divided by 3 is 1045 or we can say 3 multiplied by 1045 gives 3135. So we get 3135 is a multiple of 3. Now when z is equal to 6 then the number becomes 3165 and 3165 divided by 3 gives 1055 or we can write 3 multiplied by 1055 is equal to 3165. So we get 3165 is also a multiple of 3. Now when z is equal to 9 then the number we get is 3195 and 3195 divided by 3 gives 1065 or you can say 3 multiplied by 1065 gives 3195. So again this number is also a multiple of 3. Now since all the 4 numbers are multiples of 3 so our answer is correct. Now this is our required answer this completes the session hope you understood the solution take care and bye for now.