6. Without actual division show that 11 is a factor of each of the following numbers.
(i) 1111
(ii) 11011
(iii) 110011
(iv) 1100011
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(i) 1,111
The sum of the digits at the odd places = 1 + 1 = 2
The sum of the digits at the even places = 1 + 1 = 2
The difference of the two sums = 2 − 2 = 0
∴ 1,111 is divisible by 11 because the difference of the sums is zero.
(ii) 11,011
The sum of the digits at the odd places = 1 + 0 + 1 = 2
The sum of the digits at the even places = 1 + 1 = 2
The difference of the two sums = 2 − 2 = 0
∴ 11,011 is divisible by 11 because the difference of the sums is zero.
(iii) 1,10,011
The sum of the digits at the odd places = 1 + 0 + 1 = 2
The sum of the digits at the even places = 1 + 0 + 1 = 2
The difference of the two sums = 2 − 2 = 0
∴ 1,10,011 is divisible by 11 because the difference of the sums is zero.
(iv) 11,00,011
The sum of the digits at the odd places = 1 + 0 + 0 + 1 = 2
The sum of the digits at the even places = 1 + 0 + 1 = 2
The difference of the two sums = 2 − 2 = 0
∴ 11,00,011 is divisible by 11 because the difference of the sums is zero
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