8. Show that for all integers n, n² – 3n is even. Give a full proof supporting each statement you use and use full sentences in your answer. Note: Using statements like 'a sum of two even integers is even' without supporting algebra is not considered a full proof.

Provide explanations/justification for your answers. I have a basket with x eggs in it, and a bunch of 7-egg boxes to fill. How many boxes do I need to store all the eggs? I have a basket with x eggs in it, and a bunch of 7-egg boxes. I fill boxes one after another. The last box is not full and contains z < 7 eggs. What is z. How many bits are required to address a 2 GiB address space?

Prove the following using a direct proof. Your proof should be expressed in complete English sentences. If a, b, and c are integers such that b is a multiple of a3 and c is a multiple of b2, then c is a multiple of a6.

Exercise 4. Consider the statement VEZ, if x² is even, then x is even. Recall that we say that an integer is even if it is a multiple of 2. (a) Give the contrapositive of the above statement. (b) Give the negation of the original statement. (c) Which of the following are true? Circle all that apply. original statement Contrapositive negation Justify your answer but no proof is required.

By dragging statements from the left column to the right column below, construct a valide proof of the statement: For all integers n, if 19n is even, then n is even. The correct proof will use 3 of the statements below. Statements to choose from: Your Proof: Put chosen statements in order in this column and press the Submit Answers button. Let n be an arbitrary integer and assume n is odd. Let n be an arbitrary integer and assume 19n is even. 19n must be even Since 19 is odd and the product of odd numbers is odd, Since an even number divided by 19 must be even, 19n must be odd. n must be even Let n be an arbitrary integer and assume n is even. Since 19 is odd and the product of an odd number and an even number is even,

Consider statement P: "For all integers n, if n2 is even, then n is even." 1. Write statement P in predicate logic. Use E(x) for "x is even." 2. Write the contrapositive of statement P in predicate logic. Use O(x) for “x is odd." 3. Give a direct proof of the contrapositive of statement P. 4. In problem 1, you should have recorded a formula for P in predicate logic. Find a formula for the negation of P in predicate logic. Use rules of logic to write this statement using the 3 quantifier and the A connective, and translate this negation back into English. 5. Assume that the negation of P is true. Prove that this assumption leads to a contradiction.

1. Provide a Boolean Expression to simplify. This expression should involve an overbar over the entire expression. Present TWO different simplification proofs. The first: apply DeMorgan's law as the first step, the second: simplify under the bar first.

By dragging statements from the left column to the right column below, construct a valide proof of the statement: For all integers n, if 13n is even, then n is even. The correct proof will use 3 of the statements below. Statements to choose from: Let n be an arbitrary integer and assume n is odd. Let n be an arbitrary integer and assume n is even. Since 13 is odd and the product of an odd number and an even number is even, Let n be an arbitrary integer and assume 13n is even. 13n must be even Since 13 is odd and the product of odd numbers is odd, 13n must be odd. n must be even Since an even number divided by 13 must be even, Your Proof: Put chosen statements in order in this column and press the Submit Answers button.

5. Write the following statements symbolically and then demonstrate whether they are true or false. If true, then prove it, If false, then provide a counterexample or prove that the negation of the statement is true. (a) For all integers, if 9 divides n2, then 9 divides n. (b) There exist integers m and n such that 15m + 12n = -6.

Plz