Final answer:
The claim that n²+41n+41 is a prime number for every integer n is disproven by substituting n = 41 into the expression, resulting in a product that is clearly not prime, as it is 41 multiplied by another integer.
Step-by-step explanation:
In order to disprove the claim that n²+41n+41 is a prime number for every integer n, we can find a counterexample. Consider when n is equal to 41. Substituting n = 41 into the expression gives us 41² + 41(41) + 41, which simplifies to 41(41 + 41 + 1). Since the product of 41 and 123 is clearly not a prime number (it is 41 multiplied by another integer), this shows that the expression is not prime for every integer n. Therefore, we have disproven the claim by finding a specific integer n for which the expression is not prime. We can generalize this approach and state that for any integer n that is also a prime number, and n > 1, the expression will not yield a prime number since it will be divisible by n.