A student claims: 'If $n$ is a positive integer and $n^{2} + n$ is divisible by 6, then $n$ is divisible by 3.' Which value of $n$ is a counterexample?

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A student claims: *'For every positive integer $n$ , $n^{2} - n + 11$ is prime.'* Which value of $n$ is a counterexample?
A student claims: *'If $p$ is a prime number, then $p + 4$ is also prime.'* Which value of $p$ is a counterexample?
A student claims: *'If $x > y$ , then $x^{2} > y^{2}$ .'* Which pair $(x, y)$ is a counterexample?
A student claims: 'If a whole number is divisible by both 4 and 6, then it is divisible by 24.' Which number is a counterexample?
A student claims: *'If a real number $x$ satisfies $x^{2} > x$ , then $x > 1$ .'* Which value of $x$ is a counterexample?
A student claims: *'If $a$ and $b$ are irrational numbers, then $a + b$ is irrational.'* Which pair $(a, b)$ is a counterexample?