question_answer 23)

Write a digit in the blank space of each of the following numbers, so that the number formed is divisible by 11. (a) 92 ____ 389                                 (b) 8 ____ 9484 TIPS Firstly, assume the blank space digit as \[x\], then find the sum of odd places and even places digits (from the right) separately. Now, take difference of sum of odd places digits and sum of even places digits equal to 0 or 11 and simplify to get the value of \[x\].

Answer:

                (a) Let the required unknown digit be \[x\] Then, number be \[\begin{matrix}    9 & 2 & x & 3 & 8 & 9  \\    \downarrow  & \downarrow  & \downarrow  & \downarrow  & \downarrow  & \downarrow   \\    E & O & E & O & E & O  \\ \end{matrix}\] where, O = odd and E = even                Sum of digits at odd places from right = 9 + 3 + 2 = 14 Sum of digits at even places from right = 8 + \[x\] +9 = 17 + \[x\] \[\because \] Number is divisible by 11.                   \[\therefore \] Difference of digits will be 0 or 11. \[\Rightarrow \]               \[(17+x)-14=0\]or 11 \[\Rightarrow \]               \[17+x-14=0\] or 11 \[\Rightarrow \]\[x+3=0\]or 11 Taking difference 0, \[0,\,\,x+3=0\] \[\Rightarrow \]               \[x=03=-3\]        (not possible) Taking difference 11,      \[x=3=11\] So, required digit to write in the blank space is (b) Let the required unknown digit be \[x\]. Then, number be \[\begin{matrix}    8 & x & 9 & 4 & 8 & 4  \\    \downarrow  & \downarrow  & \downarrow  & \downarrow  & \downarrow  & \downarrow   \\    E & O & E & O & E & O  \\ \end{matrix}\] Sum of digits at odd places from right = 4 + 4 + \[x\] = 8 + \[x\] Sum of digits at even places from right = 8 + 9 + 8 = 25                 \[\because \] Number is divisible by 11.                 \[\therefore \] Difference of digits will be 0 or 11.                 \[\Rightarrow \]         \[25-(8+x)=0\]or 11                 \[\Rightarrow \]               \[258x=0\] or 11 \[\Rightarrow \] \[17x=0\] or 11 Taking difference 0, \[17-x=0\Rightarrow x=17+0~\Rightarrow x=17\]                 [but 17 is not a single digit number, so it is not possible]                 Taking difference 11, \[17-x=11\]                 \[\Rightarrow \]               \[x=1711=6\]                 So, required digit to write in the blank space is