Problem:
When 1093−93 is expressed as a single whole number, the sum of the digits is
Answer Choices:
A. 10
B. 93
C. 819
D. 826
E. 833
Solution:
Since 1093=100⋯0093 zeros, we have
−100⋯0009391 nines99⋯907
and the sum of the digits is (91×9)+7=826.
OR
Look for a pattern using simpler cases:
102−93 =103−93 =104−93 =105−93 =⋮1093−93=100−93 =1,000−93 =10,000−93 =100,000−93 =07907990799907⋮91 nines 999…907
Thus the sum of the digits is (91×9)+7=826.
Answer: D.
The problems on this page are the property of the MAA's American Mathematics Competitions