MathDB

A teacher was leading a class of four perfectly logical students. The teacher chose a set

S

of four integers and gave a different number in

S

to each student. Then the teacher announced to the class that the numbers in

S

were four consecutive two-digit positive integers, that some number in

S

was divisible by

6

, and a different number in

S

was divisible by

7

. The teacher then asked if any of the students could deduce what

S

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

S

. Find the sum of all possible values of the greatest element of

S

Problem Statement