Problem:
When Clara totaled her scores, she inadvertently reversed the units digit and the tens digit of one score. By which of the following might her incorrect sum have differed from the correct one?
Answer Choices:
A.
B.
C.
D.
E.
Solution:
Switching a digit from the units column to the tens column increases the sum by times the value of that digit. For example, switching a from the units column to the tens column increases the sum by . Similarly, switching a digit from the tens column to the units column decreases the sum by times the value of that digit. Therefore reversing two digits changes the sum by an amount that must be a multiple of . Among the given choices, only is a possible difference.
(Note: Other multiples of are also possible.)
Answer: .
The problems on this page are the property of the MAA's American Mathematics Competitions