Use a proof by contrapositive to prove that for all n E Z, n2 – 2 is not divisible by 4.

Answer:See explanation below.Step-by-step explanation:To prove by contrapositive means to assume the contrapositive of what we want to prove and it should lead us to a contradiction.So, let's assume that n²- 2  is divisible by 4Therefore there exists a k ≥1 such that n² - 2 = 4kn² = 4k + 2Case 1: Let's assume n is even.There exists a j≥1 such that n = 2j -1[tex]n^{2} = 4k +2\\(2j-1)^{2} =4k + 2\\4j^{2} -4j +1 =4k +2\\4j^{2}-4j=4k+2-1\\ 4j^{2} -4j=4k-1\\4(j^{2} -j)=4k-1\\4(j^{2} -j)+1= 4k[/tex]This would mean that 4 divides 1 which is a contradiction Case 2: Let's assume n is oddThere exists a j≥1 such that n = 2j[tex]n^{2} = 4k +2\\ (2j)^{2} =4k +2\\ 4j^{2} -2 =4k[/tex]This would mean that 4 divides -2 which is a contradiction.Therefore, we have proven that for all n ∈ Z, n²- 2 is not divisible by 4