# Liars

Level pending

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 $$xy$$.

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