question_answer 1)

Using divisibility tests, determine which of the following numbers are divisible by II? (a) 5445                                (b) 10824                             (c) 7138965                         (d) 70169308    (e) 10000001      (f) 901153 TIPS Firstly, find the difference between the sum of the digits at odd places ' (from the right) and the sum of the digits at even places (from the right) of the number. If the difference is either 0 or divisible by 11, then the number is divisible by 11, otherwise not.

Answer:

                (a) We have, 5445 \[\begin{matrix}    5 & 4 & 4 & 5  \\    \downarrow  & \downarrow  & \downarrow  & \downarrow   \\    E & O & E & O  \\ \end{matrix}\] where, O = Odd and E = Even Sum of digits at odd places from right = 5 + 4 = 9 Sum of digits at even places from right = 4 + 5 = 9 Now, difference = 9 ? 9 = 0, so 5445 is divisible by 11. (b) We have, 10824 \[\begin{matrix}    1 & 0 & 8 & 2 & 4  \\    | & | & | & | & |  \\    O & E & O & E & O  \\ \end{matrix}\] Sum of digits at odd places from right = 4 + 8 + 1 = 13 Sum of digits at even places from right = 2 + 0 = 2 Now, difference = 13 ? 2 = 11, so 10824 is divisible by 11. (c) We have, 7138965 \[\begin{matrix}    7 & 1 & 3 & 8 & 9 & 6 & 5  \\    | & | & | & | & | & | & |  \\    O & E & O & E & O & E & O  \\ \end{matrix}\] Sum of digits at odd places from right = 5 + 9 + 3 + 7 = 24 Sum of digits at even places from right = 6 + 8 + 1 = 15 Now, difference = 24 ? 15 = 9 \[\because \] 9 is not a multiple of 11, so 7138965 is not divisible by 11. (d) We have, 70169308 \[\begin{matrix}    7 & 0 & 1 & 6 & 9 & 3 & 0 & 8  \\    | & | & | & | & | & | & | & |  \\    E & O & E & O & E & O & E & O  \\ \end{matrix}\] Sum of digits at odd places from right = 8 + 3 + 6 + 0 = 17 Sum of digits at even places from right = 0 + 9 + 1 + 7 = 17 Now, difference = 17 ? 17 = 0 \[\therefore \] 70169308 is divisible by 11. (e) We have, 10000001 \[\begin{matrix}    1 & 0 & 0 & 0 & 0 & 0 & 0 & 1  \\    | & | & | & | & | & | & | & |  \\    E & O & E & O & E & O & E & O  \\ \end{matrix}\] Sum of digits at odd places from right = 1 + 0 + 0 + 0 = 1 Sum of digits at even places from right = 0 + 0 + 0 + 1 = 1 Now, difference = 1 ? 1 = 0, so 10000001 is divisible by 11. (f) We have, 901153 \[\begin{matrix}    9 & 0 & 1 & 1 & 5 & 3  \\    | & | & | & | & | & |  \\    E & O & E & O & E & O  \\ \end{matrix}\] Sum of digits at odd places from right = 3 + 1 + 0 = 4 Sum of digits at even places from right = 5 + 1 + 9 = 15 Now, difference = 15 ? 4 = 11, so 901153 is divisible by 11.