I need help with this maths problems

1. Note that 256x + 81y = 1 has the solution (xx.) = (25,-79). Solve 81x = 100 (mod 256). 2. Find the error: Theorem 1. In every set of n 2 1 people, everyone has the same name. Proof. We proceed by mathematical induction. Let P(n) be the statement "in a set of n people, everyone has the same name." In a group with just one person, everyone has the same name because there's only one person. This proves that e(1) is true. We now assume that P(n) is true, and must derive (n + 1). In a set with n + 1 people, the first n people all have the same name by P(n). Also, the last n people all have the same name by P(n). So, all n + 1 people have the same name, which is P(n). 3, Show that 2 is not a rational number. Generalize. Fix the modulus as m = 507, But I'm lazy to write "507" over and over, so I'll just write m. 4. The symbol"=" has the following properties: 1. Reflexive: 2. Symmetric: 3, Transitive 5. For arithmetic, we have the following properties: 1, negation: 2, addition: 3. subtraction: 4, multiplication: 5, cancellation: 6. The Miracle of Gauss