Problem:
In the state of Coinland, coins have values of and cents. Suppose is the value in cents of the most expensive item in Coinland that cannot be purchased using these coins with exact change. What is the sum of the digits of ?
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Solution:
The problem is asking for the sum of the digits of the greatest integer that cannot be written in the form , where , and are nonnegative integers.. If , then is a multiple of 2 . If , then is a multiple of 3 . If , then is a multiple of 5 . If none of , or equals 0 , then . Therefore prime numbers less than 31 cannot be written in the required form. In particular, 29 cannot be written in this form.
To show that 29 is the greatest positive integer that cannot be written as such a sum, observe that the following six consecutive integers can be written in this way.
Because additional 6 s can be added to each of these sums, it follows that all positive integers greater than 29 can be written in the required form. Therefore the most expensive item in Coinland that cannot be purchased using coins of the given denominations has value , and the requested digit sum is .
The problems on this page are the property of the MAA's American Mathematics Competitions