Question
Parallel Programming Task: Counting Primes using the Sieve of Eratosthenes
The Sieve of Eratosthenes (see Weisstein, nd.) is a simple algorithm for producing lists of prime numbers. you are required to implement a parallel program using MPI which uses this algorithm to evaluate the prime counting function (i.e. it determines for given N, the number of primes <= N).

2.1 Task
You are provided with a basic serial implementation of a prime number counting program, written in C++, which calculates the prime counting function for N = . Your task is to parallelize the program using MPI for running on the Neumann cluster.
You are required to:
1. Develop a parallel implementation of the algorithm using MPI, and obtain profiling data from runs on 1, 2, 4, and 8 MPI ranks. You should use the batch queue for this, rather than running interactively on the head node. In developing the parallel code, it may be helpful to profile the serial code first, to guide your choice of which parts need to be parallelized.   
2. Write a short report which describes your analysis and implementation, and which presents a critical analysis of your timing results. The report should consist of approximately 500 words (excluding references and any code appendices).

2.2 Notes on the Serial Implementation
The Sieve of Eratosthenes proceeds as follows to find all the prime numbers . Firstly, write a list of all the numbers from 2 to N. Start by crossing out all the multiples of 2 (apart from 2). Then move to the next number which is not crossed out (in this case 3). Cross out all its multiples (apart from the number itself). Repeat this process until N is reached, and the numbers which have not been crossed out then constitute a complete list of primes .
There is a significant optimization possible: since any composite number m must have a prime factor , then we can stop the crossing out process when we reach , rather than N.

The serial code provided is in two parts.
1. An initial list of primes (basePrimes)up to some is produced using the sieve. In this case, 2. A function (CountPrimes( unsigned long long int m0, unsigned long long int m1, unsigned long long int *basePrimes, int nBasePrimes) ) uses the crossing out procedure on a list of numbers from m0 to m1 to count the primes in that interval. 3. CountPrimes can be called repeatedly, summing the results, to calculate the final total required.

Notes:
I. m1 must be II. The code uses 64 bit integers (long long int) to allow for counting beyond . III. CountPrimes allocates a list of 1-byte integers to store the crossing-off status of numbers. Therefore m1 – m0 can be limited by available memory (and may cause performance problems if too large).
The serial code takes approximately 3 minutes to run on my desktop machine.
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int main(int argc, char** argv) {

    int com_sz; // size of our community or number of process
    int my_rank; // rank of the current process
   
   
    int * belowSquare; // array of number that is smaller than or equal to
                        // square root of N
    int * aboveSquare; // array of number that is larger than square root of N
    uint64_t slot_sz;    // size of the aboveSquare array per process                        
    uint64_t remainder; // the last process has to work on bigger array than
                        // the other process
                        // for example N = 18, com_sz = 2
                        // sqrt(18) = 4
                        // slot_sz = (18 - 4 - 1) / 2 = 6
                        // then process 0 will find prime number from 5 to 11
                        // "6 numbers"   
                        // process 1 find prime number from 12 to 18
                        // "7 numbers"
                        // so remainder = (18 - 4 - 1 ) % 2 = 1   
    uint64_t low;       // smallest number in the range
    uint64_t hight;    // largest number in the range
    uint64_t i, j;      // some indices
    uint64_t n;         // the largest number
    uint64_t square;    // square root of n
    vector<uint64_t> primes;    // vector of prime number that are smaller than
                        // square
    bool isPrime;       // boolean value
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      • Illuminate\Cache\Events\CacheHit (194μs)
      • Illuminate\Cache\Events\CacheHit (168μs)
      • Illuminate\Cache\Events\CacheHit (194μs)
      • Illuminate\Cache\Events\CacheHit (170μs)
      • Illuminate\Cache\Events\CacheHit (204μs)
      • Illuminate\Cache\Events\CacheHit (224μs)
      • Illuminate\Cache\Events\CacheHit (200μs)
      • Illuminate\Cache\Events\CacheHit (171μs)
      • Illuminate\Cache\Events\CacheHit (211μs)
      • Illuminate\Cache\Events\CacheHit (182μs)
      • Illuminate\Cache\Events\CacheHit (195μs)
      • Illuminate\Cache\Events\CacheHit (165μs)
      • Illuminate\Cache\Events\CacheHit (189μs)
      • Illuminate\Cache\Events\CacheHit (165μs)
      • Illuminate\Cache\Events\CacheHit (187μs)
      • Illuminate\Cache\Events\CacheHit (166μs)
      • Illuminate\Cache\Events\CacheHit (186μs)
      • Illuminate\Cache\Events\CacheHit (166μs)
      • Illuminate\Cache\Events\CacheHit (222μs)
      • Illuminate\Cache\Events\CacheHit (173μs)
      • Illuminate\Cache\Events\CacheHit (183μs)
      • Illuminate\Cache\Events\CacheHit (162μs)
      • Illuminate\Cache\Events\CacheHit (189μs)
      • Illuminate\Cache\Events\CacheHit (169μs)
      • Illuminate\Cache\Events\CacheHit (195μs)
      • Illuminate\Cache\Events\CacheHit (173μs)
      • Illuminate\Cache\Events\CacheHit (197μs)
      • Illuminate\Cache\Events\CacheHit (217μs)
      • Illuminate\Cache\Events\CacheHit (208μs)
      • Illuminate\Cache\Events\CacheHit (155μs)
      • Illuminate\Cache\Events\CacheHit (197μs)
      • Illuminate\Cache\Events\CacheHit (179μs)
      • Illuminate\Cache\Events\CacheHit (146μs)
      • Illuminate\Cache\Events\CacheHit (115μs)
      • Illuminate\Cache\Events\CacheHit (200μs)
      • Illuminate\Cache\Events\CacheHit (177μs)
      • Illuminate\Cache\Events\CacheHit (141μs)
      • Illuminate\Cache\Events\CacheHit (115μs)
      • Illuminate\Cache\Events\CacheMissed (205μs)
      • Illuminate\Database\Events\StatementPrepared (717μs)
      • Illuminate\Database\Events\QueryExecuted (1.01ms)
      • eloquent.retrieved: App\Models\SubjectCat (91μs)
      • Illuminate\Cache\Events\KeyWritten (315μs)
      • Illuminate\Cache\Events\CacheHit (173μs)
      • Illuminate\Cache\Events\CacheHit (186μs)
      • Illuminate\Cache\Events\CacheHit (290μs)
      • Illuminate\Cache\Events\CacheHit (204μs)
      • Illuminate\Cache\Events\CacheHit (197μs)
      • Illuminate\Cache\Events\CacheHit (185μs)
      • Illuminate\Cache\Events\CacheHit (166μs)
      • Illuminate\Cache\Events\CacheHit (188μs)
      • Illuminate\Cache\Events\CacheHit (167μs)
      • Illuminate\Cache\Events\CacheHit (188μs)
      • Illuminate\Cache\Events\CacheHit (170μs)
      • Illuminate\Cache\Events\CacheHit (192μs)
      • Illuminate\Cache\Events\CacheHit (165μs)
      • Illuminate\Cache\Events\CacheHit (199μs)
      • Illuminate\Cache\Events\CacheHit (169μs)
      • Illuminate\Cache\Events\CacheHit (213μs)
      • Illuminate\Cache\Events\CacheHit (171μs)
      • creating: site.layouts.app (470μs)
      • composing: site.layouts.app (22μs)
      • creating: components.canonical (394μs)
      • composing: components.canonical (84μs)
      • creating: components.open-graph (169μs)
      • composing: components.open-graph (90μs)
      • creating: site.headers.header (250μs)
      • composing: site.headers.header (77μs)
      • Illuminate\Cache\Events\CacheHit (1.69ms)
      • creating: components.footer (79μs)
      • composing: components.footer (94μs)
      • Illuminate\Cache\Events\CacheHit (829μs)
      • Illuminate\Cache\Events\CacheMissed (234μs)
      • Illuminate\Database\Events\StatementPrepared (602μs)
      • Illuminate\Database\Events\QueryExecuted (936μs)
      • eloquent.retrieved: App\Models\SubjectCat (92μs)
      • Illuminate\Cache\Events\KeyWritten (1.61ms)
      • Illuminate\Cache\Events\CacheHit (284μs)
      • Illuminate\Cache\Events\CacheHit (187μs)
      • Illuminate\Cache\Events\CacheHit (171μs)
      • Illuminate\Cache\Events\CacheHit (173μs)
      • Illuminate\Cache\Events\CacheHit (287μs)
      • Illuminate\Cache\Events\CacheHit (306μs)
      • Illuminate\Cache\Events\CacheHit (455μs)
      • Illuminate\Cache\Events\CacheHit (211μs)
      • Illuminate\Cache\Events\CacheHit (222μs)
      • Illuminate\Cache\Events\CacheHit (208μs)
      • Illuminate\Cache\Events\CacheHit (219μs)
      • Illuminate\Cache\Events\CacheHit (209μs)
      • creating: components.forms.contact-us (228μs)
      • composing: components.forms.contact-us (150μs)
      • creating: components.forms.get-started (135μs)
      • composing: components.forms.get-started (75μs)
      • creating: components.forms.free-tool-download (111μs)
      • composing: components.forms.free-tool-download (66μs)
      • creating: components.forms.claim-free-worksheet (125μs)
      • composing: components.forms.claim-free-worksheet (70μs)
      • creating: components.forms.tutor-subscription-waitlist (102μs)
      • composing: components.forms.tutor-subscription-waitlist (65μs)
      • creating: components.forms.tutor-subscription-join (100μs)
      • composing: components.forms.tutor-subscription-join (67μs)
      • creating: components.forms.tutor-support (98μs)
      • composing: components.forms.tutor-support (65μs)
      • 311 x Illuminate\Cache\Events\CacheHit (13.92%)
        66.33ms
        20 x Illuminate\Database\Events\StatementPrepared (6.94%)
        33.07ms
        20 x Illuminate\Database\Events\QueryExecuted (4.82%)
        22.96ms
        11 x Illuminate\Cache\Events\KeyWritten (1.65%)
        7.84ms
        11 x Illuminate\Cache\Events\CacheMissed (1.47%)
        7.03ms
        1 x Illuminate\Foundation\Events\LocaleUpdated (1.06%)
        5.03ms
        1 x Illuminate\Routing\Events\Routing (0.29%)
        1.39ms
        1 x Illuminate\Database\Events\ConnectionEstablished (0.25%)
        1.17ms
        12 x eloquent.retrieved: App\Models\SubjectCat (0.21%)
        1.01ms
        1 x Illuminate\Routing\Events\RouteMatched (0.12%)
        565μs
        1 x creating: site.layouts.app (0.1%)
        470μs
        1 x creating: components.canonical (0.08%)
        394μs
        1 x creating: components.breadcrumbs (0.08%)
        378μs
        1 x eloquent.booting: App\Models\SubjectCat (0.08%)
        374μs
        1 x creating: homework.show (0.06%)
        292μs
        1 x eloquent.booting: App\Models\HomeworkLibrary\HomeworkLibraryFile (0.05%)
        252μs
        1 x creating: site.headers.header (0.05%)
        250μs
        1 x eloquent.booting: App\Models\HomeworkLibrary\HomeworkLibrary (0.05%)
        241μs
        1 x eloquent.booted: App\Models\HomeworkLibrary\HomeworkLibrary (0.05%)
        237μs
        2 x eloquent.retrieved: App\Models\Subject (0.05%)
        234μs
        1 x creating: components.forms.contact-us (0.05%)
        228μs
        6 x eloquent.retrieved: App\Models\HomeworkLibrary\HomeworkLibrary (0.04%)
        203μs
        7 x eloquent.retrieved: App\Models\HomeworkLibrary\HomeworkLibraryFile (0.04%)
        181μs
        1 x creating: components.open-graph (0.04%)
        169μs
        1 x composing: homework.show (0.03%)
        150μs
        1 x composing: components.forms.contact-us (0.03%)
        150μs
        1 x creating: components.forms.get-started (0.03%)
        135μs
        1 x creating: components.forms.claim-free-worksheet (0.03%)
        125μs
        1 x composing: components.breadcrumbs (0.02%)
        114μs
        1 x creating: components.forms.free-tool-download (0.02%)
        111μs
        1 x creating: components.forms.tutor-subscription-waitlist (0.02%)
        102μs
        1 x creating: components.forms.tutor-subscription-join (0.02%)
        100μs
        1 x creating: components.forms.tutor-support (0.02%)
        98μs
        1 x composing: components.footer (0.02%)
        94μs
        1 x composing: components.open-graph (0.02%)
        90μs
        1 x composing: components.canonical (0.02%)
        84μs
        1 x creating: components.footer (0.02%)
        79μs
        1 x composing: site.headers.header (0.02%)
        77μs
        1 x composing: components.forms.get-started (0.02%)
        75μs
        1 x composing: components.forms.claim-free-worksheet (0.01%)
        70μs
        1 x composing: components.forms.tutor-subscription-join (0.01%)
        67μs
        1 x composing: components.forms.free-tool-download (0.01%)
        66μs
        1 x composing: components.forms.tutor-subscription-waitlist (0.01%)
        65μs
        1 x composing: components.forms.tutor-support (0.01%)
        65μs
        1 x eloquent.booting: App\Models\Subject (0.01%)
        54μs
        1 x eloquent.booted: App\Models\HomeworkLibrary\HomeworkLibraryFile (0.01%)
        49μs
        1 x eloquent.booted: App\Models\SubjectCat (0.01%)
        40μs
        1 x eloquent.booted: App\Models\Subject (0.01%)
        38μs
        1 x composing: site.layouts.app (0%)
        22μs
      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
      20 statements were executed, 5 of which were duplicates, 15 unique. Show only duplicated40.1ms
      • 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` = '26812' limit 1
        13.15mstwenty4_siteHomeworkLibraryController.php#97
        Bindings
        • 0: published
        • 1: 0
        • 2: 26812
        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 (228)
        1.17mstwenty4_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 (26812)
        2.29mstwenty4_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
        990μstwenty4_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` <> 26812 and `subject` = 228 and `status` = 'published' and `price` > 0 order by RAND() limit 6
        2mstwenty4_siteHomeworkLibraryRepository.php#30
        Bindings
        • 0: 26812
        • 1: 228
        • 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 (228)
        1.04mstwenty4_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` = 26812 and `solutionslibrary_files`.`solutionlib_id` is not null and `publish` = 'question' order by `order` asc, `id` asc
        1.19mstwenty4_siteHomeworkLibrary.php#260
        Bindings
        • 0: 26812
        • 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` = 26812 and `solutionslibrary_files`.`solutionlib_id` is not null and `publish` = 'teaser' order by `order` asc, `id` asc
        930μstwenty4_siteHomeworkLibrary.php#260
        Bindings
        • 0: 26812
        • 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` = 26812 and `solutionslibrary_files`.`solutionlib_id` is not null and `publish` = 'solution' order by `order` asc, `id` asc
        2.6mstwenty4_siteHomeworkLibrary.php#260
        Bindings
        • 0: 26812
        • 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
        2.57mstwenty4_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
        1.17mstwenty4_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
        1.61mstwenty4_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
        1.3mstwenty4_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
        960μstwenty4_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
        970μstwenty4_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
        1.5mstwenty4_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
        970μstwenty4_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
        1.6mstwenty4_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` = 31 limit 1
        1.14mstwenty4_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
      • select * from `subject_cats` where `subject_cats`.`id` = 36 limit 1
        950μstwenty4_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\SubjectCat
      12SubjectCat.php
      App\Models\HomeworkLibrary\HomeworkLibraryFile
      7HomeworkLibraryFile.php
      App\Models\HomeworkLibrary\HomeworkLibrary
      6HomeworkLibrary.php
      App\Models\Subject
      2Subject.php
          _token
          lx3DZVb8xSCmYwVA1UY5IcnmXnlmoZetqbLAOaGB
          utm_source
          direct
          redirectUrl
          /college-homework-library/Computer-Science/Parallel-Computing/26812
          _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/Parallel-Computing/26812
          status_code
          200
          
          status_text
          OK
          format
          html
          content_type
          text/html; charset=UTF-8
          request_query
          []
          
          request_request
          []
          
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