Problem:
Let be a -digit positive integer, such as , whose first three digits are the same as its last three digits taken in the same order. Which of the following numbers must be a factor of
Answer Choices:
A.
B.
C.
D.
E.
Solution:
Assume has the form . Then
So must be a factor of .
A positive integer is divisible by if and only if the difference of the sums of the digits in the even and odd positions in the number is divisible by . For the sum of the digits in the even positions is equal to the sum of the digits in the odd positions, so the difference of the two sums is . Hence, divides .
Answer: .
The problems on this page are the property of the MAA's American Mathematics Competitions