Question:
Without performing actual division, find the remainder when 928174653 is divided by 11.
Solution:
$928174653=$ A multiple of $11+$ (Sum of its digits at odd places - Sum of its digits at even places)
$928174653=$ A multiple of $11+\{(9+8+7+6+3)-(2+1+4+5)\}$
$928174653=$ A multiple of $11+(33-12)$
$928174653=$ A multiple of $11+21$
$928174653=$ A multiple of $11+(11 \times 1+10)$
$928174653=$ A multiple of $11+10$
Therefore, the remainder is 10
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