Write a formal proof to prove the following conjecture to be true or false. If the...

Write a formal proof to prove the following conjecture to be true or false.

If the statement is true, write a formal proof of it. If the statement is false, provide a counterexample and a slightly modified statement that is true and write a formal proof of your new statement.

Conjecture: Let w, x, y, and z be single-digit numbers. The 4-digit number wxyz* is divisible by 9 if and only if 9 divides the sum w + x + y + z.

* I don't mean the product of the of these numbers. I mean a four-digit number like 7,235 where w = 7, x = 2, y = 3, and z = 5 are the digits.

Solutions

Expert Solution

Colby Messinger answered 1 year ago

Next > < Previous

Related Solutions

Determine if the following statements are true or false. In either case, provide a formal proof...

Determine if the following statements are true or false. In either case, provide a formal proof using the definitions of the big-O, big-Omega, and big-Theta notations. For instance, to formally prove that f (n) ∈ O(g(n)) or f (n) ∉ O(g(n)), we need to demonstrate the existence of a constant c and a sufficient large n0 such that f (n) ≤ c g(n) for all n ≥ n0, or showing that there are no such values. a) 10000n2 ∈ O(n4)....

Determine whether each of the following statements is True or False. If True, write a proof....

Determine whether each of the following statements is True or False. If True, write a proof. If False, exhibit a counterexample. 1) If m, n are arbitrary positive integers, then any system of form x ≡ a (mod m) x ≡ b (mod n) has a solution. 2) If m, n are arbitrary positive integers and the system x ≡ a (mod m) x ≡ b (mod n) has a solution, then the solution is unique modulo mn. Modern Abstract...

Determine whether each of the following statements is True or False. If True, write a proof....

Contradiction proof conception Prove: If A is true, then B is true Contradiction: If A is...

Contradiction proof conception Prove: If A is true, then B is true Contradiction: If A is true, then B is false. so we suppose B is false and follow the step to prove. At the end we get if A is true then B is true so contradict our assumption However, Theorem: Let (xn) be a sequence in R. Let L∈R. If every subsequence of (xn) has a further subsequence that converges to L, then (xn) converges to L. Proof: Assume,...

1. Determine if the following statements are true or false. If a statement is true, prove...

1. Determine if the following statements are true or false. If a statement is true, prove it in general, If a statement is false, provide a specific counterexample. Let V and W be finite-dimensional vector spaces over field F, and let φ: V → W be a linear transformation. A) If φ is injective, then dim(V) ≤ dim(W). B) If dim(V) ≤ dim(W), then φ is injective. C) If φ is surjective, then dim(V) ≥ dim(W). D) If dim(V) ≥...

Write up a formal proof that the angle bisectors of a triangle are concurrent, and that...

Write up a formal proof that the angle bisectors of a triangle are concurrent, and that the point of concurrency (the incenter) is equidistant from all three sides.

Answer the following questions True or False. Write “True” or “False” in the blank. 1._________ The...

Answer the following questions True or False. Write “True” or “False” in the blank. 1._________ The rate constant changes if the concentrations of the reactant change. 2._________ If the concentration of a reactant changes, the rate of the reaction always changes. 3._________ The coefficients of the overall reaction tell us the order for each reactant. 4._________ The half-life for a first order reaction does not depend on the initial concentration. 5._________ If the rate of a reaction over a long...

True/ False Questions: For the following questions, first write whether they are true or false then...

True/ False Questions: For the following questions, first write whether they are true or false then write in a few sentences why it’s true or false. (1.5) a. The processes for producing penicillin and blue cheese both use the Penicillium fungus. b. The reproductive organs of Amanita muscaria are popular pizza toppings. c. The fungus Claviceps purpurea is thought to have triggered the infamous Salem witch trials. d. Saccharomyces cerevisiae is a common household bread mold. ...

Write TRUE if the statement is TRUE. If the statement is FALSE, write FALSE and explain...

Write TRUE if the statement is TRUE. If the statement is FALSE, write FALSE and explain why 3. The Commissioner of Internal Revenue may prescribe presumptive gross sales and receipts for a taxpayer when the latter fails to issue receipts and when he believes that the books or other records of the latter do not correctly reflect declarations in the return. 4. Jurisdiction is the power and authority of the court to hear, try, and decide a case. It can...

Determine whether each statement is true or false. If it is true, prove it. If it...

Determine whether each statement is true or false. If it is true, prove it. If it is false, give a counterexample. a) For every function f : X → Y and all A ⊆ X, we have f^−1 [f[A]] = A. (b) For every function f : X → Y and all A ⊆ X, we have f[X \ A] = Y \ f[A]. (c) For every function f : X → Y and all A, B ⊆ Y ,...

Subjects