onsider the theorem: “for any integer x, if x mod 6 = 5, then (x2-2x+4) mod 4 = 3.” (a) Give an outline for a proof of this theorem. (b) Prove this theorem.

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Activity: Prove the following mathematical statements: Write proofs for the given statements, (inserting parenthetic remarks to explain the rationale behind each step)  a) The product of an even number and any other number is even. (use direct proof)  b) If n>0 and 4n−1 is prime, then n is odd. (use contradiction or indirect proof)  c) The sum of the first n numbers is n(n+1)/2. (use mathematical induction)   Example for Direct Proof: 1. If n is even, so is n2 a. Assume n is an even number, so n = 2k ; n2 = (2k)2 b. Thus n2 = 4k2 =2(2k2); Let j = 2k2 c. Then n2 = 2j d. By definition n is even   Example for Proof by Contradiction: 1.For all real number, x and y,        if x + y ≥ 2,   then x ≥ 1 and y ≥ 1. a. Suppose the conclusion is false, that x < 1 and y < 1 b. So, (x < 1 ) + (y < 1) = x + y < 1 + 1 = 2 c. So, if x + y ≥ 2 and x + y = 2, then we come up with the realization that our claim is true   Example for Mathematical Induction: 1. 3n – 1 is a multiple of 2…