Which of the following statements is true: For all integers a, b, and c, a | be if and only if a | b and a c. O This option A number k divides the sum of three consecutive integers n, (n + 1), and (n + 2) if and only if it divides the middle integer n + 1. This option Ifrand s are integers, then r |s if and only if r | s2. O This option For all integers a, b, and c, a | (b + c) if and only if a | b and a |c

Q: Evaluate if x=4,8,3,-2 ∑x ∑x+1 ∑(x+1) 3∑x - 4

A: Given that  if x=4,8,3,-2

Q: Then A-1 If A = 5 1 -16 -3 0 0 0 -1 0 w 00+ 4 -13

A:

Q: Take any arbitrary integer n and then square it, which one is true? n2 cannot be in the form 7k + 5.…

A: Solution is given as follows :

Q: Let A be the set of odd integers, and let B be the set of positive integers less than 10. Find An B…

A:

Q: Assume a, b ER. a+b, a² + b², a³ + b³, and a + b for all integers. Prove that am + bm is an integer…

A: Let us assume that a,b∈ℝ where ℝ is the set of real numbers. It  follows that a,b∈ℤ where ℤ is the…

Q: Which of the following statements is true: For all integers a, b and c, if (a + b)|c and then a | c…

A: Here, we have to find the true statement from the given four statements. For this, we shall check…

Q: If a is any integer, is a(a + 1) even or odd?

A:

Q: Let X = {x;} be a set of n positive integers. Show that the sum of the integers of a subset of X is…

A: It is required to show that sum of elements of a subset of described set, X, is divisible by integer…

Q: Show that if a, b are real numbers then: |a| + |b| > |a + b|

A: If a,b are real numbers, then  PT a+b≥a+b Proof: Let a,b∈ℝ Let a and b denote the absolute value of…

Q: Let X be a set of integers. How many integers are needed to guarantee that there is a difference…

A: We shall be using the Pigeonhole principle

Q: Use The Fundamental Theorem of Arithmetic to show If (a, b) = 1, a|c, and b|c, then ab|c.

A:

Q: If a number is divisible by 6, then it is divisible by 2 and 3.  If a number is divisible by 2 and…

A: Solution: We can determine whether the given statements are true or false by using divisibility rule…

Q: Prove or disprove: If a and b are odd integers, then 4 | (a – b) or 4 | (a+b). -

A:

Q: Prove or disprove: For any integer a, 14 | (3a — 5) and 21 | (2a+3).

A: The given statement is FALSE.We shall provide a counterexample to show that the given statement is…