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Use the Euclidean algorithm to determine the GCD(286, 273).  Show your work. Then express the GCD(286, 273) value you identify as a linear combination of 286 and 273.  Show your work.

Prove the following statement using induction. “For all positive integers n, 3|(n3 + 2n).”  Use good proof technique.  Note:  To avoid the need for typing superscript exponents, you may use the expression ‘n^3’ to represent n3. Grading rubric:1 pt. State the basis step, then prove it.1 pt. State the inductive hypothesis.2 pt. Complete the proof of the inductive step.1 pt. State the final conclusion at the end of the proof.1 pt. Label each part: the basis step, inductive hypothesis, inductive step, and conclusion.  

Complete your choice of one of the proofs given below. PROOF 1: Prove the following statement using a proof by cases.   [Hint: there are 3 cases] “For all positive integers n with 2 ≤ n ≤ 4, n!/2 ≤ n2+1.” Use good proof technique.   Grading rubric:1 pt. State any givens and assumptions.3 pt. Clearly identify the cases and prove each case.1 pt. State the final conclusion at the end of the proof. Note:  Remember that n factorial, written as n!, is defined as n(n-1)…(2)1, the product of n times every positive integer less than n.   To avoid the need for typing superscript exponents, you may use the expression ‘2^n’ to represent 2n.  Also the ≤ symbol can be written as

Let the function f : ℕ → ℝ be defined recursively as follows:      Initial Condition:  f (0) = 18Recursive Part:  f (n + 1) = 1/9 * f (n), for n > 0 Consider how to prove the following statement about this given function f using induction. For all nonnegative integers n, f (n) = 2/9(n-1). Select the best response for each question below about how this proof by induction should be done.  Q1.  Which of the following would be a correct Basis step for this proof?   [Basis] A.  For n = 0, f(n) = f(0) = 18; also 2/9(n-1) = 2/9-1 = 2*9 = 18, so f(n) = 2/9(n-1) for n = 0. B.  For n = k, assume f(k) = 2/9(k-1) for some integer k ≥ 0, so f(n) = 2/9(n-1) for n = k. C.  For n = k+1, f(k+1) = 2/9(k+1-1) when f(k) = 2/9(k-1) for some integer k ≥ 0, so f(n) = 2/9(n-1) for n = k+1. D.  For n = 1, f(n) = f(1) = 1/9*f(0) = 1/9*18 = 2; also 2/9(n-1) = 2/90 = 2, so f(n) = 3n for n = 1. Q2.  Which of the following would be a correct Inductive Hypothesis for this proof?   [InductiveHypothesis] A.  Prove f(k) = 2/9(k-1) for some integer k ≥ 0. B.  Assume f(k+1) = 2/9(k) when f(k) = 2/9(k-1) for some integer k ≥ 0. C.  Prove f(k) = 2/9(k-1) for all integers k ≥ 0. D.  Assume f(k) = 2/9(k-1) for some integer k ≥ 0. Q3.  Which of the following would be a correct completion of the Inductive Step for this proof?   [InductiveStep] A.  f(k+1) = 1/9*f(k), which confirms the recursive part of the definition. B.  When the inductive hypothesis is true, f(k+1) = 1/9*f(k) = 1/9*2/9(k-1) = 2/9(k-1)+1 = 2/9(k) = 2/9(k+1)-1 C.  When the inductive hypothesis is true, f(k+1) = 2/9(k) = 2/9(k+1)-1 = 1/9*2/9(k-1) = 1/9*f(k), which confirms the recursive part of the definition. D.  When f(k+1) = 2/9(k+1)-1 = 2/9(k) = 1/9*2/9(k-1); also f(k+1) = 1/9*f(k), so f(k) = 2/9(k-1), confirming the induction hypothesis. Q4.  Which of the following would be a correct conclusion for this proof?   [Conclusion] A.  By the principle of mathematical induction, f(n+1) =1/9* f(n) for all integers n ≥ 0. B.  By the principle of mathematical induction, f(k) = f(k+1) for all integers k ≥ 0. C.  By the principle of mathematical induction, f(n) = 2/9(n-1) for all integers n ≥ 0. D.  By the principle of mathematical induction, f(k) = 2/9(k-1) implies f(k+1) = 2/9(k) for all integers k ≥ 0.  

What could be adjusted to improve your learning experience in this course?

The binary expansion of the decimal number 205 is ___________two. Show your work or explain your answer.

For arbitrary positive integers a, b, and m with m>1, if a(mod m) = b(mod m), then a ≡ b (mod m).

Indicate which of these listed graphs are bipartite. Select ‘True’ if the graph is bipartite; otherwise select ‘False’.  There may be more than one or none. [A]   K2 [B]   C3 [C]   Q4 [D]   W5

Consider the following problem:  1391(mod 11) = _________ Show how Fermat’s Little Theorem can be used to solve this problem. Express your answer as a non-negative integer less than the modulus. Note:  To avoid the need for typing superscript exponents, you may use the notation ‘x^n’ or the expression ‘x to the nth’ (with numbers in place of x and n), to represent xn.

Prove, or provide a counterexample to disprove, the following statement:             “The function f : ℝ ⟶ ℤ defined by f(x) = ⌊ 2x ⌋ is a bijection.” Use good proof technique.  Remember that a bijection is both one-to-one (injective) and onto (surjective).  To prove, you must demonstrate both properties are true; to disprove, you only need a counterexample that shows one of the properties is not valid. Grading rubric:1 pt.  Indicate whether you will be proving or disproving the assertion.  Also, if proving, state both definitions, one-to-one and onto; if disproving, state the definition you plan to disprove.  1 pt.  State any givens and assumptions.1 pt.  Clearly explain your reasoning.1 pt.  Remember to state the final conclusion at the end of the proof. Note:  To avoid the need for typing special symbols, instead of using the floor symbols in the function definition ⌊ 2x ⌋ you may use the expression ‘floor of ( 2x )’.