[{"data":1,"prerenderedAt":-1},["ShallowReactive",2],{"doc-detail-146441-en":3,"doc-seo-146441-105":29,"detail-sidebar-cat-0-en-105":94},{"code":4,"msg":5,"data":6},0,"success",{"doc_id":7,"user_id":8,"nickname":9,"user_avatar":10,"doc_module":4,"category_id":11,"category_name":12,"doc_title":13,"doc_description":14,"doc_content":15,"file_id":16,"file_url":17,"file_type":18,"file_size":19,"view_count":20,"is_deleted":4,"is_public":21,"is_downloadable":21,"audit_status":21,"page_count":21,"language":22,"language_code":23,"site_id":24,"html_lang":23,"table_of_contents":25,"faqs":26,"seo_title":27,"seo_description":14,"update_tm":28,"read_time":20},146441,2336474466712,"Quinn Holloway","https://ap-avatar.wpscdn.com/davatar_a8503ba1806abce46bf441b54a3ca4cd",4,"Exam","Problem Set 5","Problem Set 5 focuses on discrete mathematics and number theory tasks. It includes a classic hat-color strategy problem involving optimal coordinated guessing by a group of gnomes with limited visibility. The remaining questions address congruences, modular arithmetic, and divisibility properties, including greatest common divisor conditions, simultaneous congruences, and solving specific congruence systems. Other exercises require determining remainders from large integer division and proving implications between divisibility statements.","1. One thousand gnomes have been imprisoned by an evil wizard. The wizard informs the gnomes that the following day they will line up, facing forward, and that he will place a hat on each gnome. Each hat will be one of 41 possible colors. Starting from the back of the line, each gnome will then be permitted to guess their own hat color. After all the gnomes have 􀀌nished guessing, the gnomes that guessed correctly will be freed. Determine a strategy for the gnomes guaranteeing that at most one of them will not be freed. Each gnome can see all of the hats in front of them, and hear all of the guesses made by gnomes behind them, but they cannot see their own hat color.1  \n2. Determine the largest integer n such that n + 10 divides n3 + 100.  \n3. Suppose that a; b are two positive integers such that gcd(a; b) = 1 .  \n(a) Prove that there exists integers u; v such that the following congruences hold.  \nu 􀀑 0 (mod a) u 􀀑 1 (mod b)  \nv 􀀑 1 (mod a) v 􀀑 0 (mod b)  \n(b) Prove that for any two integers c; d, there exists an integer x such that x 􀀑 c (mod a) and x 􀀑 d (mod b) .  \n(c) Find an integer n such that n 􀀑 16 (mod 17) and n 􀀑 4 (mod 19) .  \n4. Let a; m; n be positive integers, with a 􀀕 2. Prove that if am + 1 divides an + 1, then m divides n.  \n5. Determine the remainder when 195085 is divided by 43 .  \n6. Solve the congruence x 17 􀀑 5 (mod 43) .  \n1 Don't worry: once these 999 gnomes are free they will be powerful enough to come back and rescue the last one.","cbCaifJqaUIwVQmt","https://ap.wps.com/l/cbCaifJqaUIwVQmt","pdf",85305,3,1,"English","en",105,"# Hat color strategy\n# Modular arithmetic and divisibility\n## Existence of simultaneous congruence solutions\n## Specific congruence solving\n# Remainder and congruence computation","[{\"question\":\"How do the gnomes guarantee that at most one guess will be wrong in the hat problem?\",\"answer\":\"They use a coordinated strategy based on what each gnome can observe (all hats in front and all guesses behind) to encode information so that the group can free all but possibly one gnome after the full sequence of guesses.\"},{\"question\":\"How can you find integers satisfying u ≡ 0 (mod a), u ≡ 1 (mod b) when gcd(a,b)=1?\",\"answer\":\"By using the fact that coprime moduli allow construction via modular inverses or the Chinese remainder theorem, producing integers that meet each congruence simultaneously.\"},{\"question\":\"How do you solve the congruence system n ≡ 16 (mod 17) and n ≡ 4 (mod 19)?\",\"answer\":\"Apply the Chinese remainder theorem to combine the two congruences into a single solution modulo 17·19, yielding the required integer n.\"}]","Problem Set 5 | 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do the gnomes guarantee that at most one guess will be wrong in the hat problem?","Question",{"text":78,"@type":79},"They use a coordinated strategy based on what each gnome can observe (all hats in front and all guesses behind) to encode information so that the group can free all but possibly one gnome after the full sequence of guesses.","Answer",{"name":81,"@type":76,"acceptedAnswer":82},"How can you find integers satisfying u ≡ 0 (mod a), u ≡ 1 (mod b) when gcd(a,b)=1?",{"text":83,"@type":79},"By using the fact that coprime moduli allow construction via modular inverses or the Chinese remainder theorem, producing integers that meet each congruence simultaneously.",{"name":85,"@type":76,"acceptedAnswer":86},"How do you solve the congruence system n ≡ 16 (mod 17) and n ≡ 4 (mod 19)?",{"text":87,"@type":79},"Apply the Chinese remainder theorem to combine the two congruences into a single solution modulo 17·19, yielding the required integer 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