CSCI 480 Math Exercises # 2
Prove each of the following statements. Explicitly state which method of proof you are using: direct, contrapositive, contradiction, induction, etc. Format all solutions in LATEX. Please note any time you use Sympy or other software to simplify algebraic expressions, and include the transcript of the interaction.
1. If two integers have the same parity, then their sum is even.
2. Suppose a ∈Z. If a2 is not divisible by 4, then a is odd.
3. Suppose a,b ∈Z. If 4 | (a2 + b2), then a and b are not both odd.
4. For all n ∈N: n X i=1 i(i + 1) = n(n + 1)(n + 2) 3 5. For all n ∈N: n X i=1 (8i−5) = 4n2 −n 6. For all n ∈N: 9 | (43n + 8)
