Problem:
The first three terms of a geometric sequence are the integers , and , where . What is the sum of the digits of the least possible value of ?
Answer Choices:
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Solution:
The prime factorization of 720 is . Let be the common ratio of the geometric sequence, where and are relatively prime positive integers. If had any prime factor greater than 5 , then would not be an integer. Analogously, if had any prime factor greater than 5 , then would not be an integer. It follows that , where , and are (not necessarily positive) integers. Furthermore, , and .
To minimize the value of , it suffices to minimize the value of . Taking yields the sequence . To check that no lesser values of exist, first observe that is not a possible value for , so both and are greater than 17 . This means that and must borrow at least two prime factors each from 720 , but 720 has only three distinct prime factors, so this is impossible. It follows that the least possible value of is 768 , and the requested sum of digits is .
The problems on this page are the property of the MAA's American Mathematics Competitions