 Hi and welcome to the session. Let us discuss the following question. Question says, if 3 1 z 5 is a multiple of 9, where z is a digit, what is the value of z? There are two answers for this problem. Why is this so? Now before moving on to the solution, first of all let us understand that any number n is divisible by 9 if the sum of its digits is divisible by 9. We will use this property as our key idea to solve the given question. Let us now move on to the solution. Now in the given question we are having a number 3 1 z 5 where z represents a digit and this number is a multiple of 9. Now we can say 3 1 z 5 is divisible by 9. Now we can write since 3 1 z 5 is divisible by 9, sum of its digits that is 3 plus 1 plus z plus 5 should be divisible by 9. 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 9. Really you can see here we have applied key idea. Key idea states that if any number is divisible by 9 then the sum of its digits is also divisible by 9. Now we are given in the question that 3 1 z 5 is divisible by 9. So sum of its digits is also divisible by 9. Now this is possible only when 9 plus z is equal to any multiple of 9 that is 9 plus z is equal to 9 or 18 or 27 or 36 and other such multiples. Now we know that z represents a digit so it can have value only between 0 and 9. Now if 9 plus z is equal to 9 then value of z is 0. Now 0 represents a digit so 1 possible value of z is 0. Now when 9 plus z is equal to 18 then value of z is equal to 18 minus 9 that is 9 so another value of z is 9. Now when 9 plus z is equal to 27 then value of z is given by 27 minus 9 which is equal to 18. Now 18 is not a single digit number or you can say 18 does not rise between 0 and 9 so it does not represent a digit. So this is not the possible value of z. Similarly value of z for other higher multiples is greater than 9. So possible values of z are 0 and 9. Now we can write but since z is a digit therefore 9 plus z is equal to 9 or 9 plus z is equal to 18. Now consider this equation subtracting 9 from both the sides of this equation we get value of z is equal to 9 minus 9 that is 0. So we get value of z as 0 and let us consider this equation now. Now subtracting 9 from both the sides of this equation we get value of z is equal to 18 minus 9 which is further equal to 9. So another value of z is 9. Now when value of z is 0 then the number found is 3105. Now on dividing this number by 9 we get 345 or you can say 9 multiplied by 345 is equal to 3105. So 0 is the correct value of z. Now when z is equal to 9 then the number is equal to 3195. Now on dividing 3195 by 9 we get 355. Or we can say 9 multiplied by 355 is 3195. So we get 3095 and 3195 both are multiples of 9. So our answer is correct. So this is our required answer. This completes the session hope you understood the solution take care and bye for now.