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jmc

number theory junior

Problem

Your friend has over 200 stamps. He has a stamp book where he can put 30 stamps on each page. He has placed the stamps in his stamp book in such a way that the only page which is not completely filled with stamps is the last one. The last page has 5 stamps in it. What is the smallest number of stamps your friend can have?
Solution
An integer equivalent to 5 mod 30 can be written in the form . In this situation, represents the number of stamps your friend has, while represents the number of filled pages he has. We want to solve the inequality . This inequality has solution . Since represents the number of filled pages, must be an integer. The smallest integer greater than is , so your friend has stamps.
Final answer
215