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
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