examday

/TMUA

A student calculates

for $n=0,1,2,3$ and obtains four prime numbers. The student conjectures that the expression is prime for every integer $n\geq0$.

Which statements are true?

• I. $n=10$ is a counterexample.
• II. Every positive integer $n$ that leaves remainder $0$ or $10$ when divided by $11$ is a counterexample.
• III. Every counterexample must leave remainder $0$ or $10$ when divided by $11$.