EXAMPLE: The sum of any two even integers is even. Answer: Proof. (direct) Suppose x and y are even integers. By definition of even, let x = 2k and y = 2j for some integers j and k. Then, x+y=2k+2j = 2(k + j) Note that k+j is an integer because it is the sum of integers. Therefore, by definition of even, x + y is even. (a) For any integers x, y, and z with xy, if z is divisible by x-y, then z is divisible by y-x.

Q: Let a be an integer such that a = 4 mod 12. Use the definition of mod to prove that a² = 4 mod 12-

A: Solution-:                           a≡4 mod 12a2=(4)2 mod 12     =16 mod  12     =4   mod   12

Q: write a Java program to find the Modular multiplicative inverse after taking both number from user.

A: Required:- write a Java program to find the Modular multiplicative inverse after taking both numbers…

Q: Consider the set of integers. 1. Use a direct proof to show that the sum of two even integers is…

A: 1. Use a direct proof to show that the sum of two even integers is even. Solution: let assume x and…

Q: Let P(n) be the statement that a postage of n cents can be formed using just  4-cent and 5-cent…

A: Lets see the solution of the above problem.

Q: Suppose n e Z. (That is, throughout this question, n is an arbitrary integer; you cannot pick a…

A: Given: P: If 7n+5 is odd, then n is even. Requirement: Find contrapositive of P

Q: 11. Let n is a positive integer. Prove that n is even if and only if 3n²+1 is odd. (Hint: you must…

A: Using a demonstration by contraposition, we will assume the negative of the conclusion and…

Q: Prove the following statement by contraposition. For every integer x, if 5x 2 – 2x + 1 is even, then…

A: Proof for every integer x, if 5x2 – 2x + 1 is even, then x is odd.

Q: 3. Use a truth table (either by hand or with a computer program) to prove the commutative laws for ^…

A: The commutative law for ∧ is p∧q≡q∧p lets create truth table for the law p q p∧q q∧p T T T T…

Q: Let A = {c, e,j, 1, s, u, y, z}, which of the following is true? There may'be more than one correct…

A: A={c ,e ,j ,l ,s ,u , y, z} True answers are: A. Φ ⊆ A D. {l,y,s,u} = {u,s,l,y} E. {l,y,s,u} ⊆ A F.…

Q: What will be values of AL, AH, and BL after the following piece of code is excuted? Answer in…

A: The given piece of block of code is  mov (100, AX);     ----> This instruction perform  AX = 100…

Q: (c) If x and y are integers such that x > 1, and x divides y, then xł(y+1).

A: - The question is to prove that if x and y are integers and X is greater than 1 and X divides Y…

Q: Write a formal proof of the following statement.  Be sure to copy this statement on your paper…

A: Given statement is, For all integers n, n is even if and only if 3n is even. The another implication…