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(1) For all integers and , if , then

(1) For all integers and , if  , then      True or false? [a1] (2) There exist integers  and such that   and

(1) For any integers  and , if  and , then . True or false? [a1] (2) There exist integers  and  such that ,  , and 

In this question, we will prove directly the following statement :   

(6 points for the correct answer. Extra 1 bonus point if you have correctly answered 7 questions among questions 1 through 7 and question 9.) Let 

Let  and be integers. Is true?

Suppose m and n are integers such that  and .  Then   (Please enter your answer as an integer.)

This assignment must be submitted by the end of class. Only one assignment will be submitted for each group (the same grade will be given to all those in the group). You are not to use ChatGPT or any similar AI drafting platform for this assignment, but you may use any kind of legal or internet general research sites. Please complete the names of the other students in your group:________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________

As a shareholder in Microsoft Corporation, draft a detailed activist letter critiquing the company’s recent performance and proposing constructive strategies for improvement. Your letter should reflect an understanding of Microsoft Corporation’s business landscape, incorporate stakeholder perspectives, and outline a follow-up action plan.You can look at public documents, filings and headlines and select the issue(s). You may want to choose something related to corporate governance and/or a specific incident or recent controversy involving Microsoft Corporation (e.g., labor practices, data privacy concerns, use of technology and/or AI, environmental impact etc.). Analyze the implications of this incident on the company’s reputation and performance.Assume the role of the specific stakeholder (concerned shareholder, employee, customer, or cross-over) and write your letter from this perspective.Incorporate relevant data and metrics to support your critique. You can limit your research to one or two key performance indicators (“KPIs”) such as financial performance, market share statistics, customer satisfaction scores, etc. that support your arguments.If relevant, you can identify a competitor of Microsoft Corporation and discuss what Microsoft Corporation can learn from their strategy.You can consider addressing how your recommendations would benefit the chosen stakeholder group and improve corporate responsibility.You can reflect briefly on the ethical implications of your recommendations and consider how they align with corporate social responsibility and the long-term sustainability of Amazon’s business practices.Propose 2-3 actionable recommendations with a short explanation of potential challenges.

A(n) ________________________ is needed in a sales message when the audience is not known to have expressed an interest in a product, idea, or service previously.