Problem:
Let equal the sum of the digits of positive integer . For example, . For a particular positive integer . Which of the following could be the value of ?
Answer Choices:
A.
B.
C.
D.
E.
Solution:
Note that unless the numeral for ends with a . Moreover, if the numeral for ends with exactly s, then . Thus the possible values of when are all of the form , where . Of the choices, only can be formed in this manner, and will equal if, for example, consists of consecutive s preceded by s.
The value of a positive integer is congruent to the sum of its digits modulo . Therefore , so . Of the given choices, only meets this requirement.
The problems on this page are the property of the MAA's American Mathematics Competitions