Question
PROBLEM 1
A ternary sequence is like a binary sequence, only with 0, 1 and 2 as possible digits instead of solely 0 and 1. Find recurrence relations and initial conditions that describe the number of n-digit ternary sequences with:
(A) no occurrence of the subsequence “210”.
(B) an even number of 0s.
Solution.

PROBLEM 2
Find functions fitting the following recurrences, and prove your answers by induction.
(A) f(1) = 1, f(n) = 2f(n/2) + n
(B) g(1) = 1, g(n) = g(n/2) + n/2
Solution.

PROBLEM 3
At one time, the Guinness Book of World Records reported that the “greatest human calculator” was a guy who could compute 13th roots of 100-digit numbers that were powers of 13. What a curious choice of tasks!
(A) Prove that d¹³ ≡ d mod 10for 0 ≤ d < 10
(B) Now prove that n¹³ ≡ n mod 10 for all n.
Solution.

PROBLEM 4
You may have heard of Fermat’s Last Theorem, conjectured by Pierre de Fermat in 1637 and famously proven by Andrew Wiles in the 1990s. Fermat also proved another result: “Fermat’s Little Theorem,”, which states that if p is a prime number, and a is an integer not divisible by p, then a p−1 ≡ 1 (mod p). We can prove Fermat’s Little Theorem using induction on a.
(A) Check that the base case, a = 1, holds.
(B) Assume that a p ≡ a (mod p). Expand (a + 1)p using the binomial theorem, then use induction to conclude that a p ≡ a (mod p) for all a.
(C) Use the fact that a p ≡ a (mod p) to establish Fermat’s Little Theorem.
Solution.
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      • Illuminate\Cache\Events\CacheHit (125μs)
      • Illuminate\Cache\Events\CacheHit (100μs)
      • Illuminate\Cache\Events\CacheHit (121μs)
      • Illuminate\Cache\Events\CacheHit (96μs)
      • Illuminate\Cache\Events\CacheHit (109μs)
      • Illuminate\Cache\Events\CacheHit (93μs)
      • Illuminate\Cache\Events\CacheHit (112μs)
      • Illuminate\Cache\Events\CacheHit (100μs)
      • Illuminate\Cache\Events\CacheHit (124μs)
      • Illuminate\Cache\Events\CacheHit (100μs)
      • Illuminate\Cache\Events\CacheHit (115μs)
      • Illuminate\Cache\Events\CacheHit (99μs)
      • Illuminate\Cache\Events\CacheHit (158μs)
      • Illuminate\Cache\Events\CacheHit (105μs)
      • Illuminate\Cache\Events\CacheHit (118μs)
      • Illuminate\Cache\Events\CacheHit (99μs)
      • Illuminate\Cache\Events\CacheHit (114μs)
      • Illuminate\Cache\Events\CacheHit (102μs)
      • Illuminate\Cache\Events\CacheHit (117μs)
      • Illuminate\Cache\Events\CacheHit (108μs)
      • Illuminate\Cache\Events\CacheHit (128μs)
      • Illuminate\Cache\Events\CacheHit (105μs)
      • Illuminate\Cache\Events\CacheHit (119μs)
      • Illuminate\Cache\Events\CacheHit (107μs)
      • Illuminate\Cache\Events\CacheHit (167μs)
      • Illuminate\Cache\Events\CacheHit (120μs)
      • Illuminate\Cache\Events\CacheHit (129μs)
      • Illuminate\Cache\Events\CacheHit (105μs)
      • Illuminate\Cache\Events\CacheHit (118μs)
      • Illuminate\Cache\Events\CacheHit (99μs)
      • creating: site.layouts.app (374μs)
      • composing: site.layouts.app (17μs)
      • creating: components.canonical (297μs)
      • composing: components.canonical (70μs)
      • creating: components.open-graph (120μs)
      • composing: components.open-graph (49μs)
      • creating: site.headers.header (169μs)
      • composing: site.headers.header (50μs)
      • Illuminate\Cache\Events\CacheHit (1.14ms)
      • creating: components.footer (68μs)
      • composing: components.footer (69μs)
      • Illuminate\Cache\Events\CacheHit (598μs)
      • Illuminate\Cache\Events\CacheMissed (182μs)
      • Illuminate\Database\Events\StatementPrepared (736μs)
      • Illuminate\Database\Events\QueryExecuted (926μs)
      • eloquent.retrieved: App\Models\SubjectCat (81μs)
      • Illuminate\Cache\Events\KeyWritten (341μs)
      • Illuminate\Cache\Events\CacheHit (296μs)
      • Illuminate\Cache\Events\CacheHit (159μs)
      • Illuminate\Cache\Events\CacheHit (132μs)
      • Illuminate\Cache\Events\CacheHit (137μs)
      • Illuminate\Cache\Events\CacheHit (149μs)
      • Illuminate\Cache\Events\CacheHit (240μs)
      • Illuminate\Cache\Events\CacheHit (232μs)
      • Illuminate\Cache\Events\CacheHit (126μs)
      • Illuminate\Cache\Events\CacheHit (137μs)
      • Illuminate\Cache\Events\CacheHit (123μs)
      • Illuminate\Cache\Events\CacheHit (125μs)
      • Illuminate\Cache\Events\CacheHit (112μs)
      • creating: components.forms.contact-us (309μs)
      • composing: components.forms.contact-us (126μs)
      • creating: components.forms.get-started (128μs)
      • composing: components.forms.get-started (47μs)
      • creating: components.forms.free-tool-download (87μs)
      • composing: components.forms.free-tool-download (41μs)
      • creating: components.forms.claim-free-worksheet (286μs)
      • composing: components.forms.claim-free-worksheet (59μs)
      • creating: components.forms.tutor-subscription-waitlist (107μs)
      • composing: components.forms.tutor-subscription-waitlist (41μs)
      • creating: components.forms.tutor-subscription-join (83μs)
      • composing: components.forms.tutor-subscription-join (38μs)
      • creating: components.forms.tutor-support (79μs)
      • composing: components.forms.tutor-support (41μs)
      • 312 x Illuminate\Cache\Events\CacheHit (17.08%)
        60.35ms
        19 x Illuminate\Database\Events\StatementPrepared (10.08%)
        35.60ms
        19 x Illuminate\Database\Events\QueryExecuted (9.34%)
        33.01ms
        10 x Illuminate\Cache\Events\KeyWritten (1.73%)
        6.12ms
        10 x Illuminate\Cache\Events\CacheMissed (1.72%)
        6.09ms
        1 x Illuminate\Foundation\Events\LocaleUpdated (1.35%)
        4.77ms
        1 x Illuminate\Routing\Events\Routing (0.29%)
        1.03ms
        1 x Illuminate\Database\Events\ConnectionEstablished (0.26%)
        926μs
        11 x eloquent.retrieved: App\Models\SubjectCat (0.24%)
        849μs
        1 x Illuminate\Routing\Events\RouteMatched (0.14%)
        481μs
        1 x creating: site.layouts.app (0.11%)
        374μs
        1 x eloquent.booting: App\Models\SubjectCat (0.1%)
        352μs
        1 x creating: components.forms.contact-us (0.09%)
        309μs
        1 x creating: components.canonical (0.08%)
        297μs
        7 x eloquent.retrieved: App\Models\HomeworkLibrary\HomeworkLibrary (0.08%)
        290μs
        1 x creating: components.forms.claim-free-worksheet (0.08%)
        286μs
        1 x creating: homework.show (0.08%)
        275μs
        11 x eloquent.retrieved: App\Models\HomeworkLibrary\HomeworkLibraryFile (0.08%)
        268μs
        1 x creating: components.breadcrumbs (0.07%)
        264μs
        2 x eloquent.retrieved: App\Models\Subject (0.06%)
        226μs
        1 x eloquent.booting: App\Models\HomeworkLibrary\HomeworkLibrary (0.05%)
        190μs
        1 x creating: site.headers.header (0.05%)
        169μs
        1 x eloquent.booted: App\Models\HomeworkLibrary\HomeworkLibrary (0.05%)
        165μs
        1 x composing: components.breadcrumbs (0.04%)
        133μs
        1 x creating: components.forms.get-started (0.04%)
        128μs
        1 x composing: components.forms.contact-us (0.04%)
        126μs
        1 x composing: homework.show (0.03%)
        123μs
        1 x creating: components.open-graph (0.03%)
        120μs
        1 x eloquent.booting: App\Models\HomeworkLibrary\HomeworkLibraryFile (0.03%)
        116μs
        1 x creating: components.forms.tutor-subscription-waitlist (0.03%)
        107μs
        1 x creating: components.forms.free-tool-download (0.02%)
        87μs
        1 x creating: components.forms.tutor-subscription-join (0.02%)
        83μs
        1 x creating: components.forms.tutor-support (0.02%)
        79μs
        1 x composing: components.canonical (0.02%)
        70μs
        1 x composing: components.footer (0.02%)
        69μs
        1 x creating: components.footer (0.02%)
        68μs
        1 x composing: components.forms.claim-free-worksheet (0.02%)
        59μs
        1 x eloquent.booting: App\Models\Subject (0.02%)
        56μs
        1 x eloquent.booted: App\Models\SubjectCat (0.02%)
        56μs
        1 x composing: site.headers.header (0.01%)
        50μs
        1 x composing: components.open-graph (0.01%)
        49μs
        1 x composing: components.forms.get-started (0.01%)
        47μs
        1 x eloquent.booted: App\Models\HomeworkLibrary\HomeworkLibraryFile (0.01%)
        44μs
        1 x composing: components.forms.free-tool-download (0.01%)
        41μs
        1 x composing: components.forms.tutor-subscription-waitlist (0.01%)
        41μs
        1 x composing: components.forms.tutor-support (0.01%)
        41μs
        1 x composing: components.forms.tutor-subscription-join (0.01%)
        38μs
        1 x eloquent.booted: App\Models\Subject (0.01%)
        38μs
        1 x composing: site.layouts.app (0%)
        17μs
      14 templates were rendered
      • 1x homework.showshow.blade.phpblade
      • 1x components.breadcrumbsbreadcrumbs.blade.phpblade
      • 1x site.layouts.appapp.blade.phpblade
      • 1x components.canonicalcanonical.blade.phpblade
      • 1x components.open-graphopen-graph.blade.phpblade
      • 1x site.headers.headerheader.blade.phpblade
      • 1x components.footerfooter.blade.phpblade
      • 1x components.forms.contact-uscontact-us.blade.phpblade
      • 1x components.forms.get-startedget-started.blade.phpblade
      • 1x components.forms.free-tool-downloadfree-tool-download.blade.phpblade
      • 1x components.forms.claim-free-worksheetclaim-free-worksheet.blade.phpblade
      • 1x components.forms.tutor-subscription-waitlisttutor-subscription-waitlist.blade.phpblade
      • 1x components.forms.tutor-subscription-jointutor-subscription-join.blade.phpblade
      • 1x components.forms.tutor-supporttutor-support.blade.phpblade
      uri
      GET college-homework-library/{category}/{subject}/{id}
      middleware
      web, utm.parameters
      controller
      App\Http\Controllers\HomeworkLibraryController@show
      namespace
      where
      as
      homework.show
      file
      app/Http/Controllers/HomeworkLibraryController.php:79-176
      19 statements were executed, 5 of which were duplicates, 14 unique. Show only duplicated53.89ms
      • Connection Establishedtwenty4_siteHomeworkLibraryController.php#91
        Backtrace
        • 13. app/Http/Controllers/HomeworkLibraryController.php:91
        • 14. vendor/laravel/framework/src/Illuminate/Routing/Controller.php:54
        • 15. vendor/laravel/framework/src/Illuminate/Routing/ControllerDispatcher.php:43
        • 16. vendor/laravel/framework/src/Illuminate/Routing/Route.php:260
        • 17. vendor/laravel/framework/src/Illuminate/Routing/Route.php:205
      • select * from `solutionslibrary` where `status` = 'published' and `price` > 0 and `solutionslibrary`.`id` = '22940' limit 1
        21.02mstwenty4_siteHomeworkLibraryController.php#97
        Bindings
        • 0: published
        • 1: 0
        • 2: 22940
        Backtrace
        • 16. app/Http/Controllers/HomeworkLibraryController.php:97
        • 17. vendor/laravel/framework/src/Illuminate/Routing/Controller.php:54
        • 18. vendor/laravel/framework/src/Illuminate/Routing/ControllerDispatcher.php:43
        • 19. vendor/laravel/framework/src/Illuminate/Routing/Route.php:260
        • 20. vendor/laravel/framework/src/Illuminate/Routing/Route.php:205
      • select * from `subjects` where `subjects`.`id` in (15)
        900μstwenty4_siteHomeworkLibraryController.php#97
        Backtrace
        • 21. app/Http/Controllers/HomeworkLibraryController.php:97
        • 22. vendor/laravel/framework/src/Illuminate/Routing/Controller.php:54
        • 23. vendor/laravel/framework/src/Illuminate/Routing/ControllerDispatcher.php:43
        • 24. vendor/laravel/framework/src/Illuminate/Routing/Route.php:260
        • 25. vendor/laravel/framework/src/Illuminate/Routing/Route.php:205
      • select * from `solutionslibrary_files` where `solutionslibrary_files`.`solutionlib_id` in (22940)
        2.86mstwenty4_siteHomeworkLibraryController.php#97
        Backtrace
        • 21. app/Http/Controllers/HomeworkLibraryController.php:97
        • 22. vendor/laravel/framework/src/Illuminate/Routing/Controller.php:54
        • 23. vendor/laravel/framework/src/Illuminate/Routing/ControllerDispatcher.php:43
        • 24. vendor/laravel/framework/src/Illuminate/Routing/Route.php:260
        • 25. vendor/laravel/framework/src/Illuminate/Routing/Route.php:205
      • select * from `subject_cats` where `subject_cats`.`id` = 3 limit 1
        1.22mstwenty4_siteHomeworkLibrary.php#201
        Bindings
        • 0: 3
        Backtrace
        • 20. app/Models/HomeworkLibrary/HomeworkLibrary.php:201
        • 26. app/Http/Controllers/HomeworkLibraryController.php:105
        • 27. vendor/laravel/framework/src/Illuminate/Routing/Controller.php:54
        • 28. vendor/laravel/framework/src/Illuminate/Routing/ControllerDispatcher.php:43
        • 29. vendor/laravel/framework/src/Illuminate/Routing/Route.php:260
      • select * from `solutionslibrary` where `id` <> 22940 and `subject` = 15 and `status` = 'published' and `price` > 0 order by RAND() limit 6
        10.82mstwenty4_siteHomeworkLibraryRepository.php#30
        Bindings
        • 0: 22940
        • 1: 15
        • 2: published
        • 3: 0
        Backtrace
        • 14. app/Repositories/HomeworkLibraryRepository.php:30
        • 15. vendor/laravel/framework/src/Illuminate/Cache/Repository.php:397
        • 16. vendor/laravel/framework/src/Illuminate/Cache/CacheManager.php:419
        • 18. app/Repositories/HomeworkLibraryRepository.php:39
        • 19. app/Http/Controllers/HomeworkLibraryController.php:139
      • select * from `subjects` where `subjects`.`id` in (15)
        980μstwenty4_siteHomeworkLibraryRepository.php#30
        Backtrace
        • 19. app/Repositories/HomeworkLibraryRepository.php:30
        • 20. vendor/laravel/framework/src/Illuminate/Cache/Repository.php:397
        • 21. vendor/laravel/framework/src/Illuminate/Cache/CacheManager.php:419
        • 23. app/Repositories/HomeworkLibraryRepository.php:39
        • 24. app/Http/Controllers/HomeworkLibraryController.php:139
      • select * from `solutionslibrary_files` where `solutionslibrary_files`.`solutionlib_id` = 22940 and `solutionslibrary_files`.`solutionlib_id` is not null and `publish` = 'question' order by `order` asc, `id` asc
        1.21mstwenty4_siteHomeworkLibrary.php#260
        Bindings
        • 0: 22940
        • 1: question
        Backtrace
        • 15. app/Models/HomeworkLibrary/HomeworkLibrary.php:260
        • 16. app/Transformers/HomeworkLibrary/HomeworkLibraryTransformer.php:58
        • 19. vendor/league/fractal/src/TransformerAbstract.php:128
        • 20. vendor/league/fractal/src/TransformerAbstract.php:107
        • 21. vendor/league/fractal/src/Scope.php:383
      • select * from `solutionslibrary_files` where `solutionslibrary_files`.`solutionlib_id` = 22940 and `solutionslibrary_files`.`solutionlib_id` is not null and `publish` = 'teaser' order by `order` asc, `id` asc
        1.12mstwenty4_siteHomeworkLibrary.php#260
        Bindings
        • 0: 22940
        • 1: teaser
        Backtrace
        • 15. app/Models/HomeworkLibrary/HomeworkLibrary.php:260
        • 16. app/Transformers/HomeworkLibrary/HomeworkLibraryTransformer.php:69
        • 19. vendor/league/fractal/src/TransformerAbstract.php:128
        • 20. vendor/league/fractal/src/TransformerAbstract.php:107
        • 21. vendor/league/fractal/src/Scope.php:383
      • select * from `solutionslibrary_files` where `solutionslibrary_files`.`solutionlib_id` = 22940 and `solutionslibrary_files`.`solutionlib_id` is not null and `publish` = 'solution' order by `order` asc, `id` asc
        930μstwenty4_siteHomeworkLibrary.php#260
        Bindings
        • 0: 22940
        • 1: solution
        Backtrace
        • 15. app/Models/HomeworkLibrary/HomeworkLibrary.php:260
        • 16. app/Transformers/HomeworkLibrary/HomeworkLibraryTransformer.php:80
        • 19. vendor/league/fractal/src/TransformerAbstract.php:128
        • 20. vendor/league/fractal/src/TransformerAbstract.php:107
        • 21. vendor/league/fractal/src/Scope.php:383
      • select * from `subject_cats` where `subject_cats`.`id` = 3 limit 1
        1.12mstwenty4_siteHomeworkLibrary.php#201
        Bindings
        • 0: 3
        Backtrace
        • 20. app/Models/HomeworkLibrary/HomeworkLibrary.php:201
        • 32. vendor/laravel/framework/src/Illuminate/Filesystem/Filesystem.php:110
        • 33. vendor/laravel/framework/src/Illuminate/View/Engines/PhpEngine.php:58
        • 34. vendor/livewire/livewire/src/ComponentConcerns/RendersLivewireComponents.php:69
        • 35. vendor/laravel/framework/src/Illuminate/View/Engines/CompilerEngine.php:70
      • select * from `subject_cats` where `subject_cats`.`id` = 1 limit 1
        860μstwenty4_siteSubject.php#100
        Bindings
        • 0: 1
        Backtrace
        • 18. app/Models/Subject.php:100
        • 19. vendor/laravel/framework/src/Illuminate/Cache/Repository.php:397
        • 20. vendor/laravel/framework/src/Illuminate/Cache/CacheManager.php:419
        • 22. app/Models/Subject.php:101
        • 34. vendor/laravel/framework/src/Illuminate/Filesystem/Filesystem.php:110
      • select * from `subject_cats` where `subject_cats`.`id` = 3 limit 1
        940μstwenty4_siteSubject.php#100
        Bindings
        • 0: 3
        Backtrace
        • 18. app/Models/Subject.php:100
        • 19. vendor/laravel/framework/src/Illuminate/Cache/Repository.php:397
        • 20. vendor/laravel/framework/src/Illuminate/Cache/CacheManager.php:419
        • 22. app/Models/Subject.php:101
        • 34. vendor/laravel/framework/src/Illuminate/Filesystem/Filesystem.php:110
      • select * from `subject_cats` where `subject_cats`.`id` = 4 limit 1
        840μstwenty4_siteSubject.php#100
        Bindings
        • 0: 4
        Backtrace
        • 18. app/Models/Subject.php:100
        • 19. vendor/laravel/framework/src/Illuminate/Cache/Repository.php:397
        • 20. vendor/laravel/framework/src/Illuminate/Cache/CacheManager.php:419
        • 22. app/Models/Subject.php:101
        • 34. vendor/laravel/framework/src/Illuminate/Filesystem/Filesystem.php:110
      • select * from `subject_cats` where `subject_cats`.`id` = 10 limit 1
        970μstwenty4_siteSubject.php#100
        Bindings
        • 0: 10
        Backtrace
        • 18. app/Models/Subject.php:100
        • 19. vendor/laravel/framework/src/Illuminate/Cache/Repository.php:397
        • 20. vendor/laravel/framework/src/Illuminate/Cache/CacheManager.php:419
        • 22. app/Models/Subject.php:101
        • 34. vendor/laravel/framework/src/Illuminate/Filesystem/Filesystem.php:110
      • select * from `subject_cats` where `subject_cats`.`id` = 33 limit 1
        1.15mstwenty4_siteSubject.php#100
        Bindings
        • 0: 33
        Backtrace
        • 18. app/Models/Subject.php:100
        • 19. vendor/laravel/framework/src/Illuminate/Cache/Repository.php:397
        • 20. vendor/laravel/framework/src/Illuminate/Cache/CacheManager.php:419
        • 22. app/Models/Subject.php:101
        • 34. vendor/laravel/framework/src/Illuminate/Filesystem/Filesystem.php:110
      • select * from `subject_cats` where `subject_cats`.`id` = 11 limit 1
        1.08mstwenty4_siteSubject.php#100
        Bindings
        • 0: 11
        Backtrace
        • 18. app/Models/Subject.php:100
        • 19. vendor/laravel/framework/src/Illuminate/Cache/Repository.php:397
        • 20. vendor/laravel/framework/src/Illuminate/Cache/CacheManager.php:419
        • 22. app/Models/Subject.php:101
        • 34. vendor/laravel/framework/src/Illuminate/Filesystem/Filesystem.php:110
      • select * from `subject_cats` where `subject_cats`.`id` = 5 limit 1
        870μstwenty4_siteSubject.php#100
        Bindings
        • 0: 5
        Backtrace
        • 18. app/Models/Subject.php:100
        • 19. vendor/laravel/framework/src/Illuminate/Cache/Repository.php:397
        • 20. vendor/laravel/framework/src/Illuminate/Cache/CacheManager.php:419
        • 22. app/Models/Subject.php:101
        • 34. vendor/laravel/framework/src/Illuminate/Filesystem/Filesystem.php:110
      • select * from `subject_cats` where `subject_cats`.`id` = 34 limit 1
        3.89mstwenty4_siteSubject.php#100
        Bindings
        • 0: 34
        Backtrace
        • 18. app/Models/Subject.php:100
        • 19. vendor/laravel/framework/src/Illuminate/Cache/Repository.php:397
        • 20. vendor/laravel/framework/src/Illuminate/Cache/CacheManager.php:419
        • 22. app/Models/Subject.php:101
        • 34. vendor/laravel/framework/src/Illuminate/Filesystem/Filesystem.php:110
      • select * from `subject_cats` where `subject_cats`.`id` = 36 limit 1
        1.11mstwenty4_siteSubject.php#100
        Bindings
        • 0: 36
        Backtrace
        • 18. app/Models/Subject.php:100
        • 19. vendor/laravel/framework/src/Illuminate/Cache/Repository.php:397
        • 20. vendor/laravel/framework/src/Illuminate/Cache/CacheManager.php:419
        • 22. app/Models/Subject.php:101
        • 28. view::components.footer:170
      App\Models\HomeworkLibrary\HomeworkLibraryFile
      11HomeworkLibraryFile.php
      App\Models\SubjectCat
      11SubjectCat.php
      App\Models\HomeworkLibrary\HomeworkLibrary
      7HomeworkLibrary.php
      App\Models\Subject
      2Subject.php
          _token
          GK2DJIVLbfvHZkuB1sSWriZ6C03FA3tXyb1FaY28
          utm_source
          direct
          redirectUrl
          /college-homework-library/Computer-Science/Discrete-Math/22940
          _previous
          array:1 [ "url" => "https://staging.dev.24houranswers.com/college-homework-library/Computer-Scienc...
          _flash
          array:2 [ "old" => [] "new" => [] ]
          PHPDEBUGBAR_STACK_DATA
          []
          path_info
          /college-homework-library/Computer-Science/Discrete-Math/22940
          status_code
          200
          
          status_text
          OK
          format
          html
          content_type
          text/html; charset=UTF-8
          request_query
          []
          
          request_request
          []
          
          request_headers
          0 of 0
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          request_cookies
          []
          
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