Problem:
Find the number of five-digit positive integers, , that satisfy the following conditions:
(a) the number is divisible by ,
(b) the first and last digits of are equal, and
(c) the sum of the digits of is divisible by .
Solution:
If a number satisfies the given conditions, then because its first and last digits are equal, and the number is a multiple of , its first and last digits must be . The sum of the middle three digits must also be a multiple of . If one chooses any two values for the first two of those three digits, there will be two possible choices for the third digit that will make the sum equal to a multiple of . Thus there are choices for such a number.
The problems on this page are the property of the MAA's American Mathematics Competitions