Quadratic residues of Z ∗. We can also consider the quadratic res...

Question
Quadratic residues of Z ∗.
We can also consider the quadratic residues of the additive group Zn (i.e. any element y ∈ Zn is a quadratic residue if and only if there exists an x ∈ Zn with 2x = y mod n.).
(a) What are the quadratic residues in Zp for p an odd prime?
(b) Let n = pq be a product of two odd primes p and q. What are the quadratic residues in Zn ?
(c) Let n be an even integer. What are the quadratic residues in Zn?
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      • Illuminate\Cache\Events\CacheHit (163μs)
      • creating: site.layouts.app (477μs)
      • composing: site.layouts.app (17μs)
      • creating: components.canonical (313μs)
      • composing: components.canonical (107μs)
      • creating: components.open-graph (194μs)
      • composing: components.open-graph (53μs)
      • creating: site.headers.header (310μs)
      • composing: site.headers.header (80μs)
      • Illuminate\Cache\Events\CacheHit (1.93ms)
      • creating: components.footer (70μs)
      • composing: components.footer (90μs)
      • Illuminate\Cache\Events\CacheHit (906μs)
      • Illuminate\Cache\Events\CacheHit (374μs)
      • Illuminate\Cache\Events\CacheHit (251μs)
      • Illuminate\Cache\Events\CacheHit (179μs)
      • Illuminate\Cache\Events\CacheHit (212μs)
      • Illuminate\Cache\Events\CacheHit (178μs)
      • Illuminate\Cache\Events\CacheHit (178μs)
      • Illuminate\Cache\Events\CacheHit (300μs)
      • Illuminate\Cache\Events\CacheHit (330μs)
      • Illuminate\Cache\Events\CacheHit (217μs)
      • Illuminate\Cache\Events\CacheHit (216μs)
      • Illuminate\Cache\Events\CacheHit (219μs)
      • Illuminate\Cache\Events\CacheHit (320μs)
      • Illuminate\Cache\Events\CacheHit (331μs)
      • creating: components.forms.contact-us (226μs)
      • composing: components.forms.contact-us (157μs)
      • creating: components.forms.get-started (106μs)
      • composing: components.forms.get-started (74μs)
      • creating: components.forms.free-tool-download (99μs)
      • composing: components.forms.free-tool-download (64μs)
      • creating: components.forms.claim-free-worksheet (116μs)
      • composing: components.forms.claim-free-worksheet (68μs)
      • creating: components.forms.tutor-subscription-waitlist (100μs)
      • composing: components.forms.tutor-subscription-waitlist (61μs)
      • creating: components.forms.tutor-subscription-join (99μs)
      • composing: components.forms.tutor-subscription-join (65μs)
      • creating: components.forms.tutor-support (109μs)
      • composing: components.forms.tutor-support (81μs)
      • 314 x Illuminate\Cache\Events\CacheHit (15.48%)
        83.04ms
        17 x Illuminate\Database\Events\StatementPrepared (6.29%)
        33.74ms
        17 x Illuminate\Database\Events\QueryExecuted (4.68%)
        25.09ms
        8 x Illuminate\Cache\Events\CacheMissed (1.53%)
        8.22ms
        1 x Illuminate\Foundation\Events\LocaleUpdated (0.79%)
        4.26ms
        8 x Illuminate\Cache\Events\KeyWritten (0.76%)
        4.08ms
        1 x Illuminate\Routing\Events\Routing (0.25%)
        1.32ms
        2 x eloquent.retrieved: App\Models\Subject (0.24%)
        1.30ms
        1 x Illuminate\Database\Events\ConnectionEstablished (0.18%)
        972μs
        9 x eloquent.retrieved: App\Models\SubjectCat (0.14%)
        778μs
        1 x Illuminate\Routing\Events\RouteMatched (0.09%)
        507μs
        1 x creating: site.layouts.app (0.09%)
        477μs
        6 x eloquent.retrieved: App\Models\HomeworkLibrary\HomeworkLibraryFile (0.07%)
        391μs
        1 x eloquent.booting: App\Models\SubjectCat (0.07%)
        389μs
        1 x creating: homework.show (0.07%)
        385μs
        1 x creating: components.breadcrumbs (0.07%)
        357μs
        1 x creating: components.canonical (0.06%)
        313μs
        1 x creating: site.headers.header (0.06%)
        310μs
        7 x eloquent.retrieved: App\Models\HomeworkLibrary\HomeworkLibrary (0.06%)
        299μs
        1 x creating: components.forms.contact-us (0.04%)
        226μs
        1 x eloquent.booting: App\Models\HomeworkLibrary\HomeworkLibrary (0.04%)
        208μs
        1 x eloquent.booted: App\Models\HomeworkLibrary\HomeworkLibrary (0.04%)
        204μs
        1 x creating: components.open-graph (0.04%)
        194μs
        1 x composing: components.forms.contact-us (0.03%)
        157μs
        1 x composing: homework.show (0.03%)
        153μs
        1 x composing: components.breadcrumbs (0.03%)
        146μs
        1 x eloquent.booting: App\Models\Subject (0.02%)
        128μs
        1 x creating: components.forms.claim-free-worksheet (0.02%)
        116μs
        1 x eloquent.booting: App\Models\HomeworkLibrary\HomeworkLibraryFile (0.02%)
        109μs
        1 x creating: components.forms.tutor-support (0.02%)
        109μs
        1 x composing: components.canonical (0.02%)
        107μs
        1 x creating: components.forms.get-started (0.02%)
        106μs
        1 x creating: components.forms.tutor-subscription-waitlist (0.02%)
        100μs
        1 x creating: components.forms.free-tool-download (0.02%)
        99μs
        1 x creating: components.forms.tutor-subscription-join (0.02%)
        99μs
        1 x composing: components.footer (0.02%)
        90μs
        1 x composing: components.forms.tutor-support (0.02%)
        81μs
        1 x composing: site.headers.header (0.01%)
        80μs
        1 x composing: components.forms.get-started (0.01%)
        74μs
        1 x creating: components.footer (0.01%)
        70μs
        1 x composing: components.forms.claim-free-worksheet (0.01%)
        68μs
        1 x composing: components.forms.tutor-subscription-join (0.01%)
        65μs
        1 x composing: components.forms.free-tool-download (0.01%)
        64μs
        1 x composing: components.forms.tutor-subscription-waitlist (0.01%)
        61μs
        1 x eloquent.booted: App\Models\Subject (0.01%)
        60μs
        1 x eloquent.booted: App\Models\HomeworkLibrary\HomeworkLibraryFile (0.01%)
        58μs
        1 x composing: components.open-graph (0.01%)
        53μs
        1 x eloquent.booted: App\Models\SubjectCat (0.01%)
        43μs
        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
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      app/Http/Controllers/HomeworkLibraryController.php:79-176
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      • 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` = '6148' limit 1
        19.12mstwenty4_siteHomeworkLibraryController.php#97
        Bindings
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        • 1: 0
        • 2: 6148
        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 (34)
        1.13mstwenty4_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 (6148)
        1.01mstwenty4_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` = 1 limit 1
        1.7mstwenty4_siteHomeworkLibrary.php#201
        Bindings
        • 0: 1
        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` <> 6148 and `subject` = 34 and `status` = 'published' and `price` > 0 order by RAND() limit 6
        7.97mstwenty4_siteHomeworkLibraryRepository.php#30
        Bindings
        • 0: 6148
        • 1: 34
        • 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 (34)
        1.11mstwenty4_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` = 6148 and `solutionslibrary_files`.`solutionlib_id` is not null and `publish` = 'question' order by `order` asc, `id` asc
        1mstwenty4_siteHomeworkLibrary.php#260
        Bindings
        • 0: 6148
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        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` = 6148 and `solutionslibrary_files`.`solutionlib_id` is not null and `publish` = 'teaser' order by `order` asc, `id` asc
        1.15mstwenty4_siteHomeworkLibrary.php#260
        Bindings
        • 0: 6148
        • 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` = 6148 and `solutionslibrary_files`.`solutionlib_id` is not null and `publish` = 'solution' order by `order` asc, `id` asc
        1.03mstwenty4_siteHomeworkLibrary.php#260
        Bindings
        • 0: 6148
        • 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` = 1 limit 1
        890μstwenty4_siteHomeworkLibrary.php#201
        Bindings
        • 0: 1
        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
        1.03mstwenty4_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
        990μstwenty4_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
        1.02mstwenty4_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
        850μstwenty4_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
        1.12mstwenty4_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
        930μstwenty4_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` = 31 limit 1
        930μstwenty4_siteSubject.php#100
        Bindings
        • 0: 31
        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
      App\Models\SubjectCat
      9SubjectCat.php
      App\Models\HomeworkLibrary\HomeworkLibrary
      7HomeworkLibrary.php
      App\Models\HomeworkLibrary\HomeworkLibraryFile
      6HomeworkLibraryFile.php
      App\Models\Subject
      2Subject.php
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          status_text
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          request_cookies
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