# Liars

In a classroom, the teacher announces a prime number to a class of 10 students.The 10 students then each make one statement, in the following order:

Student 1: The number is greater that 1000

Student 2: Student 1 is telling the truth.

Student 3: If the number is greater than 1000000, Student 1 is lying.

Student 4: If the number is odd, Student 2 is lying.

Student 5: The number is greater than 10000000.

Student 6: The sum of the digits of the number is 27.

Student 7: If Student 4 is telling the truth, Student 6 is lying.

Student 8: The number can be expressed in the form \(2^n-1\) for an integer \(n\).

Student 9: Student 3 is telling the truth if and only if Student 7 is lying.

Student 10: The number can be expressed as the sum of two perfect squares.

Let \(x\) be the minimum number of students who could be telling the truth, and let \(y\) be the maximum. Evaluate \(xy\).