Three university students, A, B and C, applied for a job in the computer laboratory. All were straight-A students in both math and computer science. Since there was more than a month before the school year started, I devised a test to choose the best candidate.

On Day 1 I met with the 3 students and carefully explained the procedure. I had chosen a certain Pythagorean right triangle with integer sides. The hypotenuse was between 300 and 2300, inclusive. I would secretly tell A the length of the shorter leg, B the length of the longer leg, and C the length of the hypotenuse. Then I would ask them, in turn, if they could tell me the lengths of all 3 sides, giving them a full day each time to consider their answers. They would all hear the other students' answers.

On Day 2 I assembled the 3 students and asked A what are the lengths of the 3 sides. A did not know.
On Day 3 I assembled them and asked B for the lengths of the 3 sides. B did not know.
On Day 4 I asked C for the lengths of the 3 sides. C did not know.
On Day 5 I asked A again for the lengths of the 3 sides. A did not know.
I continued asking them in turn for lengths of the 3 sides. Finally, on Day 17, after I had asked A for the sixth time, and A did not know, both B and C shouted out "I know!"

What is the length of the shortest side?