Problem:
A teacher was leading a class of four perfectly logical students. The teacher chose a set of four integers and gave a different number in to each student. Then the teacher announced to the class that the numbers in were four consecutive two-digit positive integers, that some number in was divisible by , and a different number in was divisible by . The teacher then asked if any of the students could deduce what is, but in unison, all of the students replied no. However, upon hearing that all four students replied no, each student was able to determine the elements of . Find the sum of all possible values of the greatest element of .
Solution:
Answer (258):
Let and denote the two numbers from the set that are divisible by 6 and 7 , respectively. If a student holds the number , then that student knows that the other numbers belong to the interval . For any integer , there is only one multiple of 7 in that interval. Therefore after the announcement, all four students knew the value of .
If contained or , then, after the announcement, at least one student would have known the values in . Thus, upon hearing that no student could determine the elements of , all of the students knew that the values in lay in the interval . There is at most one multiple of 6 in that interval. Therefore, upon hearing that no student could determine the elements of , all of the students also knew the value of .
If and are not consecutive, then is either or , and after the announcement, all students know because the other possible multiples of or , respectively, are too far from . Then because at least one of or is the least (or greatest) number among those four integers, the student who had the greatest (or least) number would be able to deduce all four integers. Because nobody knew the answer immediately after the announcement, and must be consecutive integers, and they must be the middle two numbers from the set . That is why all the students knew the elements of after it was clear that no student could use the information in the announcement to deduce the elements of .
The requested sum is .
The problems and solutions on this page are the property of the MAA's American Mathematics Competitions