Test the divisibility of each of the following numbers by 11:

Solution:

A given number is divisible by 11, if the difference between the sum of its digits at odd places and the sum of its digits at even places is either 0 or a number divisible by 11.

(i) 22222

For the given number,

sum of the digits at odd places = 2 + 2 + 2 = 6

sum of digits at even places = 2 + 2 = 4

Difference of the above sums = 6 − 4 = 2

Since the difference is not 0 and neither a number divisible by 11 so, the 22222 is not divisible by 11.

(ii) 444444

For the given number,

sum of the digits at odd places = 4 + 4 + 4 = 12

sum of digits at even places = 4 + 4 + 4 = 12

Difference of the above sums = 12 − 12 = 0

Since the difference is 0 so the given number 444444 is divisible by 11.

(iii) 379654

For the given number,

sum of the digits at odd places = 4 + 6 + 7 = 17

sum of digits at even places = 5 + 9 + 3 = 17

Difference of the above sums = 17 − 17 = 0

Since the difference is 0 so the given number 379654 is divisible by 11.

(iv) 1057982

For the given number,

sum of the digits at odd places = 2 + 9 + 5 + 1 = 17

sum of digits at even places = 8 + 7 + 0 = 15

Difference of the above sums = 17 − 15= 2

Since the difference is not 0 and neither a number divisible by 11 so, the 1057982 is not divisible by 11.

(v) 6543207

For the given number,

sum of the digits at odd places = 7 + 2 + 4 + 6 = 19

sum of digits at even places = 0 + 3 + 5 = 8

Difference of the above sums = 19 − 8 = 11

Since the difference is 11 which is surely divisible by 11 so the given number 6543207 is divisible by 11.

(vi) 818532

For the given number,

sum of the digits at odd places = 2 + 5 + 1 = 8

sum of digits at even places = 3 + 8 + 8 = 19

Difference of the above sums = 19 − 8 = 11

Since the difference is 11 which is surely divisible by 11 so the given number 818532 is divisible by 11.

(vii) 900163

For the given number,

sum of the digits at odd places = 3 + 1 + 0 = 4

sum of digits at even places = 6 + 0 + 9 = 15

Difference of the above sums = 15 − 4 = 11

Since the difference is 11 which is surely divisible by 11 so the given number 900163 is divisible by 11.

(viii) 7531622

For the given number,

sum of the digits at odd places = 2 + 6 + 3 + 7 = 18

sum of digits at even places = 2 + 1 + 5 = 8

Difference of the above sums = 18 − 8 = 10

Since the difference is 10 which is not divisible by 11 so the given number 7531622 is not divisible by 11.