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Number Theory

The Fibonacci Numbers

Number theory deals with the analysis of integers, i.e. whole numbers without fractions or decimals. Many applications of modeling are by nature only valid for whole numbers and can never be parts of a number. For example, when counting people, or animals. They are ordered on the numbering line ranging from negative infinity, ... -3, -2, -1, 0, 1, 2, … up to positive infinity.

One important application of number theory is in the Fibonacci Numbers, presented originally in the 13^th century in Liber Abaci ^[1]. This is widely regarded as a catalyst in bringing a more modern understanding of numbers, introducing the Arabic numbering system to Europe ^[2]. The system was formulated by Italian mathematics professor Leonardo Pisano Fibonacci, a mathematician from the Italian city of Pisa and widely renowned as one of the greatest mathematicians of the Medieval era ^{[3] [4]}.

Fibonacci studied rabbit populations and looked at a simplified model of the reproduction formulated as: ^[5]

Rabbits grow and can produce offspring after one month;
Each female produces two new offspring (one male, one female) each month;
No rabbits ever die.

This lead to the world famous equation: ^[6]

This may seem rather moderate at first sight, but leads to very high numbers as the index n grows:

n	Fn
10	55
30	832040
50	12586269025
100	354224848179261915075
300	222232244629420445529739893461909967206666939096499764990979600

Table 1. Some Fibonacci Numbers^[7]

Although the Fibonacci numbers were formulated on a very idealized and simple model, they have relevance in many forms of combinatorics problems including stock trading, art, computer science and biology.

References

1. Fibonacci, L.P.: Liber Abaci. 1202.

2. The Life and Numbers of Fibonacci. Plus Maths, https://plus.maths.org/content/life -and -numbers fibonacci

3. Who was Fibonacci? Ron Knott, Surrey University, UK. http://www.maths.surrey.ac.uk/hosted -sites/R.Knott/Fibonacci/fibBio.html

4. Devlin, K.: The Man of Numbers, Fibonaccis Arithmetic Revolution, Walker and Company 2011.

5. Fibonacci and the rabbits the story. The Fibonacci Project, University of Bayreuth, Germany, http://fibonacci.uni -de/project/fibonacci -and -the -rabbits/the -story.html

6. Fibonacci Number, Wolfram Mathworld, http://mathworld.wolfram.com/FibonacciNumber.html

7. The First 2000 Fibonacci Numbers. The Online Encyclopedia of Integer Sequences, https://oeis.org/A000045/b000045.txt

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Our tutors are just as dedicated to your success in class as you are, so they are available around the clock to assist you with questions, homework, exam preparation and any Number Theory related assignments you need extra help completing.

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1 x creating: site.blocks.how-it-works (0.02%)	293μs
1 x creating: site.blocks.text-with-image (0.02%)	283μs
4 x composing: components.faq-item (0.02%)	267μs
1 x creating: components.forms.contact-us (0.02%)	254μs
1 x creating: site.headers.header (0.02%)	244μs
2 x eloquent.retrieved: App\Models\Subject (0.02%)	241μs
1 x eloquent.booting: App\Models\Twill\Slugs\TwillFaqSlug (0.02%)	239μs
2 x eloquent.retrieved: App\Models\SubjectCat (0.02%)	233μs
6 x eloquent.retrieved: App\Models\User\Tutor (0.02%)	227μs
1 x creating: components.breadcrumbs (0.02%)	226μs
1 x creating: site.layouts.app (0.02%)	224μs
1 x eloquent.booting: App\Models\Subject (0.02%)	220μs
1 x eloquent.booted: App\Models\User\Tutor (0.02%)	214μs
6 x eloquent.retrieved: App\Models\HomeworkLibrary\HomeworkLibrary (0.02%)	213μs
1 x composing: components.forms.contact-us (0.02%)	204μs
1 x eloquent.booting: App\Models\User\Tutor (0.02%)	203μs
1 x creating: components.tutors-cards (0.02%)	199μs
1 x eloquent.booting: App\Models\Meta (0.02%)	184μs
1 x creating: site.blocks.work-with-tutors (0.01%)	163μs
1 x eloquent.booted: App\Models\Twill\TwillFaq (0.01%)	159μs
1 x creating: components.open-graph (0.01%)	148μs
1 x eloquent.booted: App\Models\HomeworkLibrary\HomeworkLibrary (0.01%)	140μs
1 x creating: site.blocks.submit-homework (0.01%)	138μs
1 x eloquent.booting: App\Models\HomeworkLibrary\HomeworkLibrary (0.01%)	134μs
1 x creating: components.forms.get-started (0.01%)	134μs
1 x eloquent.booting: App\Models\Twill\TwillFaq (0.01%)	125μs
1 x composing: components.breadcrumbs (0.01%)	120μs
1 x creating: subjects.show (0.01%)	116μs
1 x composing: components.homework-blocks (0.01%)	115μs
1 x creating: site.blocks.subject-tutors (0.01%)	105μs
1 x composing: components.tutors-cards (0.01%)	105μs
1 x composing: subjects.show (0.01%)	104μs
1 x creating: components.forms.free-tool-download (0.01%)	99μs
1 x creating: components.forms.claim-free-worksheet (0.01%)	98μs
1 x composing: components.canonical (0.01%)	97μs
1 x composing: components.footer (0.01%)	96μs
1 x creating: components.forms.tutor-subscription-waitlist (0.01%)	95μs
1 x creating: components.forms.tutor-support (0.01%)	94μs
1 x creating: components.forms.tutor-subscription-join (0.01%)	94μs
1 x creating: components.footer (0.01%)	90μs
1 x creating: site.blocks.faq (0.01%)	88μs
1 x eloquent.booted: App\Models\Twill\Slugs\TwillFaqSlug (0.01%)	87μs
1 x composing: components.forms.free-tool-download (0.01%)	83μs
1 x eloquent.booting: App\Models\SubjectCat (0.01%)	77μs
1 x composing: components.forms.tutor-support (0.01%)	76μs
1 x eloquent.retrieved: App\Models\Meta (0.01%)	73μs
1 x composing: components.forms.get-started (0.01%)	72μs
1 x composing: components.forms.claim-free-worksheet (0.01%)	71μs
1 x composing: site.headers.header (0.01%)	70μs
1 x composing: components.open-graph (0.01%)	69μs
1 x composing: components.forms.tutor-subscription-join (0.01%)	67μs
1 x composing: components.forms.tutor-subscription-waitlist (0.01%)	66μs
1 x eloquent.booted: App\Models\Subject (0.01%)	65μs
1 x eloquent.booted: App\Models\User\TutorAttributes\TutorDegree (0%)	59μs
1 x eloquent.booted: App\Models\Meta (0%)	56μs
1 x eloquent.booted: App\Models\SubjectCat (0%)	53μs
1 x composing: site.blocks.submit-homework (0%)	30μs
1 x composing: site.blocks.subject-homework (0%)	25μs
1 x composing: site.blocks.reviews (0%)	23μs
1 x composing: site.blocks.how-it-works (0%)	22μs
1 x composing: site.blocks.text-with-image (0%)	20μs
1 x composing: site.layouts.app (0%)	20μs
1 x composing: site.blocks.subject-tutors (0%)	18μs
1 x composing: site.blocks.faq (0%)	18μs
1 x composing: site.blocks.work-with-tutors (0%)	17μs

34 templates were rendered

1x subjects.showshow.blade.phpblade
1x components.breadcrumbsbreadcrumbs.blade.phpblade
1x site.blocks.how-it-workshow-it-works.blade.phpblade
1x site.blocks.subject-tutorssubject-tutors.blade.phpblade
1x components.tutors-cardstutors-cards.blade.phpblade
6x components.tutors-cardtutors-card.blade.phpblade
1x site.blocks.subject-homeworksubject-homework.blade.phpblade
1x components.homework-blockshomework-blocks.blade.phpblade
1x site.blocks.reviewsreviews.blade.phpblade
1x site.blocks.submit-homeworksubmit-homework.blade.phpblade
1x site.blocks.faqfaq.blade.phpblade
4x components.faq-itemfaq-item.blade.phpblade
1x site.blocks.text-with-imagetext-with-image.blade.phpblade
1x site.blocks.work-with-tutorswork-with-tutors.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 subjects/{categoryUri}/{subjectUri?}
middleware: web, utm.parameters
controller: App\Http\Controllers\SubjectController@show
namespace
where
as: subjects.show
file: app/Http/Controllers/SubjectController.php:43-96

23 statements were executed, 2 of which were duplicates, 21 unique. Show only duplicated115ms

Connection Establishedtwenty4_siteSubjectRepository.php#123

Backtrace

13. app/Repositories/SubjectRepository.php:123
14. app/Http/Controllers/SubjectController.php:54
15. vendor/laravel/framework/src/Illuminate/Routing/Controller.php:54
16. vendor/laravel/framework/src/Illuminate/Routing/ControllerDispatcher.php:43
17. vendor/laravel/framework/src/Illuminate/Routing/Route.php:260

select * from `subjects` where `urlFriendlyTitle` = 'Number-Theory' limit 1

5.71mstwenty4_siteSubjectRepository.php#123

Bindings	0: Number-Theory
Backtrace	15. app/Repositories/SubjectRepository.php:123 16. app/Http/Controllers/SubjectController.php:54 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

select * from `subject_cats` where `subject_cats`.`id` in (1)

880μstwenty4_siteSubjectRepository.php#123

Backtrace

20. app/Repositories/SubjectRepository.php:123
21. app/Http/Controllers/SubjectController.php:54
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

select * from `meta` where `meta`.`id` = 3 limit 1

860μstwenty4_siteMetaService.php#118

Bindings	0: 3
Backtrace	19. app/Services/MetaService.php:118 20. app/Services/MetaService.php:108 21. app/Services/MetaService.php:81 22. app/Http/Controllers/SubjectController.php:89 23. vendor/laravel/framework/src/Illuminate/Routing/Controller.php:54

Connection Establishedtwenty4_siteTutorRepository.php#40

Backtrace

13. app/Repositories/TutorRepository.php:40
14. app/View/Components/TutorsCards.php:28
15. app/View/Components/TutorsCards.php:22
17. vendor/laravel/framework/src/Illuminate/Container/Container.php:929
18. vendor/laravel/framework/src/Illuminate/Container/Container.php:770

select count(*) as aggregate from `users` where (`subjectsQualified` like '%|68|%' or `DSF_subjectsQualified` like '%|68|%' or `allsubjects` = 1) and `disable_profile` = 0 and `picture` != '' and `solutionrequestQualified` = 1 and `is_suspended` = 0 and `is_deactivated` = 0 and `status` != '' and `status` = 'tutor'

13.32mstwenty4_siteTutorsCards.php#35

Bindings	0: %\|68\|% 1: %\|68\|% 2: 1 3: 0 4: 5: 1 6: 0 7: 0 8: 9: tutor
Backtrace	15. app/View/Components/TutorsCards.php:35 16. app/View/Components/TutorsCards.php:22 18. vendor/laravel/framework/src/Illuminate/Container/Container.php:929 19. vendor/laravel/framework/src/Illuminate/Container/Container.php:770 20. vendor/laravel/framework/src/Illuminate/Foundation/Application.php:933

select * from `users` where (`subjectsQualified` like '%|68|%' or `DSF_subjectsQualified` like '%|68|%' or `allsubjects` = 1) and `disable_profile` = 0 and `picture` != '' and `solutionrequestQualified` = 1 and `is_suspended` = 0 and `is_deactivated` = 0 and `status` != '' and `status` = 'tutor' order by `completed_requests` desc limit 6 offset 0

13.22mstwenty4_siteTutorsCards.php#35

Bindings	0: %\|68\|% 1: %\|68\|% 2: 1 3: 0 4: 5: 1 6: 0 7: 0 8: 9: tutor
Backtrace	15. app/View/Components/TutorsCards.php:35 16. app/View/Components/TutorsCards.php:22 18. vendor/laravel/framework/src/Illuminate/Container/Container.php:929 19. vendor/laravel/framework/src/Illuminate/Container/Container.php:770 20. vendor/laravel/framework/src/Illuminate/Foundation/Application.php:933

select * from `tutor_degrees` where `tutor_degrees`.`tutor_id` = 6002 and `tutor_degrees`.`tutor_id` is not null order by `year` desc limit 1

1.03mstwenty4_siteTutor.php#361

Bindings	0: 6002
Backtrace	18. app/Models/User/Tutor.php:361 19. view::components.tutors-card:11 21. vendor/laravel/framework/src/Illuminate/Filesystem/Filesystem.php:110 22. vendor/laravel/framework/src/Illuminate/View/Engines/PhpEngine.php:58 23. vendor/livewire/livewire/src/ComponentConcerns/RendersLivewireComponents.php:69

select * from `tutor_degrees` where `tutor_degrees`.`tutor_id` = 2821 and `tutor_degrees`.`tutor_id` is not null order by `year` desc limit 1

3.35mstwenty4_siteTutor.php#361

Bindings	0: 2821
Backtrace	18. app/Models/User/Tutor.php:361 19. view::components.tutors-card:11 21. vendor/laravel/framework/src/Illuminate/Filesystem/Filesystem.php:110 22. vendor/laravel/framework/src/Illuminate/View/Engines/PhpEngine.php:58 23. vendor/livewire/livewire/src/ComponentConcerns/RendersLivewireComponents.php:69

select * from `tutor_degrees` where `tutor_degrees`.`tutor_id` = 6531 and `tutor_degrees`.`tutor_id` is not null order by `year` desc limit 1

5.67mstwenty4_siteTutor.php#361

Bindings	0: 6531
Backtrace	18. app/Models/User/Tutor.php:361 19. view::components.tutors-card:11 21. vendor/laravel/framework/src/Illuminate/Filesystem/Filesystem.php:110 22. vendor/laravel/framework/src/Illuminate/View/Engines/PhpEngine.php:58 23. vendor/livewire/livewire/src/ComponentConcerns/RendersLivewireComponents.php:69

select * from `tutor_degrees` where `tutor_degrees`.`tutor_id` = 10818 and `tutor_degrees`.`tutor_id` is not null order by `year` desc limit 1

14.5mstwenty4_siteTutor.php#361

Bindings	0: 10818
Backtrace	18. app/Models/User/Tutor.php:361 19. view::components.tutors-card:11 21. vendor/laravel/framework/src/Illuminate/Filesystem/Filesystem.php:110 22. vendor/laravel/framework/src/Illuminate/View/Engines/PhpEngine.php:58 23. vendor/livewire/livewire/src/ComponentConcerns/RendersLivewireComponents.php:69

select * from `tutor_degrees` where `tutor_degrees`.`tutor_id` = 76351 and `tutor_degrees`.`tutor_id` is not null order by `year` desc limit 1

1.77mstwenty4_siteTutor.php#361

Bindings	0: 76351
Backtrace	18. app/Models/User/Tutor.php:361 19. view::components.tutors-card:11 21. vendor/laravel/framework/src/Illuminate/Filesystem/Filesystem.php:110 22. vendor/laravel/framework/src/Illuminate/View/Engines/PhpEngine.php:58 23. vendor/livewire/livewire/src/ComponentConcerns/RendersLivewireComponents.php:69

select * from `tutor_degrees` where `tutor_degrees`.`tutor_id` = 69722 and `tutor_degrees`.`tutor_id` is not null order by `year` desc limit 1

2.4mstwenty4_siteTutor.php#361

Bindings	0: 69722
Backtrace	18. app/Models/User/Tutor.php:361 19. view::components.tutors-card:11 21. vendor/laravel/framework/src/Illuminate/Filesystem/Filesystem.php:110 22. vendor/laravel/framework/src/Illuminate/View/Engines/PhpEngine.php:58 23. vendor/livewire/livewire/src/ComponentConcerns/RendersLivewireComponents.php:69

select count(*) as aggregate from `solutionslibrary` where exists (select * from `subjects` where `solutionslibrary`.`subject` = `subjects`.`id`) and `subject` = 68 and `status` = 'published' and `price` > 0

13.2mstwenty4_siteHomeworkBlocks.php#27

Bindings	0: 68 1: published 2: 0
Backtrace	15. app/View/Components/HomeworkBlocks.php:27 16. vendor/laravel/framework/src/Illuminate/View/Component.php:102 17. view::site.blocks.subject-homework:4 19. vendor/laravel/framework/src/Illuminate/Filesystem/Filesystem.php:110 20. vendor/laravel/framework/src/Illuminate/View/Engines/PhpEngine.php:58

select * from `solutionslibrary` where exists (select * from `subjects` where `solutionslibrary`.`subject` = `subjects`.`id`) and `subject` = 68 and `status` = 'published' and `price` > 0 order by `purchase_count` desc limit 6 offset 0

1.27mstwenty4_siteHomeworkBlocks.php#27

Bindings	0: 68 1: published 2: 0
Backtrace	15. app/View/Components/HomeworkBlocks.php:27 16. vendor/laravel/framework/src/Illuminate/View/Component.php:102 17. view::site.blocks.subject-homework:4 19. vendor/laravel/framework/src/Illuminate/Filesystem/Filesystem.php:110 20. vendor/laravel/framework/src/Illuminate/View/Engines/PhpEngine.php:58

select * from `subjects` where `subjects`.`id` in (68)

5.46mstwenty4_siteHomeworkBlocks.php#27

Backtrace

20. app/View/Components/HomeworkBlocks.php:27
21. vendor/laravel/framework/src/Illuminate/View/Component.php:102
22. view::site.blocks.subject-homework:4
24. vendor/laravel/framework/src/Illuminate/Filesystem/Filesystem.php:110
25. vendor/laravel/framework/src/Illuminate/View/Engines/PhpEngine.php:58

select * from `subject_cats` where `subject_cats`.`id` in (1)

7.31mstwenty4_siteHomeworkBlocks.php#27

Backtrace

25. app/View/Components/HomeworkBlocks.php:27
26. vendor/laravel/framework/src/Illuminate/View/Component.php:102
27. view::site.blocks.subject-homework:4
29. vendor/laravel/framework/src/Illuminate/Filesystem/Filesystem.php:110
30. vendor/laravel/framework/src/Illuminate/View/Engines/PhpEngine.php:58

select * from `twill_faqs` where exists (select * from `twill_faq_slugs` where `twill_faqs`.`id` = `twill_faq_slugs`.`twill_faq_id` and `slug` = 'can-you-help-me-with-my-homework-in-less-than-24-hours' and `active` = 1 and `locale` = 'en' and `twill_faq_slugs`.`deleted_at` is null) and `twill_faqs`.`deleted_at` is null limit 1

1.08mstwenty4_siteFaqItem.php#15

Bindings	0: can-you-help-me-with-my-homework-in-less-than-24-hours 1: 1 2: en
Backtrace	15. app/View/Components/FaqItem.php:15 16. vendor/laravel/framework/src/Illuminate/View/Component.php:102 17. view::site.blocks.faq:7 19. vendor/laravel/framework/src/Illuminate/Filesystem/Filesystem.php:110 20. vendor/laravel/framework/src/Illuminate/View/Engines/PhpEngine.php:58

select * from `twill_faq_slugs` where `twill_faq_slugs`.`twill_faq_id` in (1) and `twill_faq_slugs`.`deleted_at` is null

950μstwenty4_siteFaqItem.php#15

Backtrace

20. app/View/Components/FaqItem.php:15
21. vendor/laravel/framework/src/Illuminate/View/Component.php:102
22. view::site.blocks.faq:7
24. vendor/laravel/framework/src/Illuminate/Filesystem/Filesystem.php:110
25. vendor/laravel/framework/src/Illuminate/View/Engines/PhpEngine.php:58

select * from `twill_faqs` where exists (select * from `twill_faq_slugs` where `twill_faqs`.`id` = `twill_faq_slugs`.`twill_faq_id` and `slug` = 'can-you-help-me-with-my-examquiztest' and `active` = 1 and `locale` = 'en' and `twill_faq_slugs`.`deleted_at` is null) and `twill_faqs`.`deleted_at` is null limit 1

980μstwenty4_siteFaqItem.php#15

Bindings	0: can-you-help-me-with-my-examquiztest 1: 1 2: en
Backtrace	15. app/View/Components/FaqItem.php:15 16. vendor/laravel/framework/src/Illuminate/View/Component.php:102 17. view::site.blocks.faq:24 19. vendor/laravel/framework/src/Illuminate/Filesystem/Filesystem.php:110 20. vendor/laravel/framework/src/Illuminate/View/Engines/PhpEngine.php:58

select * from `twill_faq_slugs` where `twill_faq_slugs`.`twill_faq_id` in (2) and `twill_faq_slugs`.`deleted_at` is null

870μstwenty4_siteFaqItem.php#15

Backtrace

20. app/View/Components/FaqItem.php:15
21. vendor/laravel/framework/src/Illuminate/View/Component.php:102
22. view::site.blocks.faq:24
24. vendor/laravel/framework/src/Illuminate/Filesystem/Filesystem.php:110
25. vendor/laravel/framework/src/Illuminate/View/Engines/PhpEngine.php:58

select * from `twill_faqs` where exists (select * from `twill_faq_slugs` where `twill_faqs`.`id` = `twill_faq_slugs`.`twill_faq_id` and `slug` = 'how-much-will-it-cost' and `active` = 1 and `locale` = 'en' and `twill_faq_slugs`.`deleted_at` is null) and `twill_faqs`.`deleted_at` is null limit 1

8.84mstwenty4_siteFaqItem.php#15

Bindings	0: how-much-will-it-cost 1: 1 2: en
Backtrace	15. app/View/Components/FaqItem.php:15 16. vendor/laravel/framework/src/Illuminate/View/Component.php:102 17. view::site.blocks.faq:41 19. vendor/laravel/framework/src/Illuminate/Filesystem/Filesystem.php:110 20. vendor/laravel/framework/src/Illuminate/View/Engines/PhpEngine.php:58

select * from `twill_faq_slugs` where `twill_faq_slugs`.`twill_faq_id` in (3) and `twill_faq_slugs`.`deleted_at` is null

6.97mstwenty4_siteFaqItem.php#15

Backtrace

20. app/View/Components/FaqItem.php:15
21. vendor/laravel/framework/src/Illuminate/View/Component.php:102
22. view::site.blocks.faq:41
24. vendor/laravel/framework/src/Illuminate/Filesystem/Filesystem.php:110
25. vendor/laravel/framework/src/Illuminate/View/Engines/PhpEngine.php:58

select * from `twill_faqs` where exists (select * from `twill_faq_slugs` where `twill_faqs`.`id` = `twill_faq_slugs`.`twill_faq_id` and `slug` = 'what-kind-of-payments-do-you-accept' and `active` = 1 and `locale` = 'en' and `twill_faq_slugs`.`deleted_at` is null) and `twill_faqs`.`deleted_at` is null limit 1

920μstwenty4_siteFaqItem.php#15

Bindings	0: what-kind-of-payments-do-you-accept 1: 1 2: en
Backtrace	15. app/View/Components/FaqItem.php:15 16. vendor/laravel/framework/src/Illuminate/View/Component.php:102 17. view::site.blocks.faq:58 19. vendor/laravel/framework/src/Illuminate/Filesystem/Filesystem.php:110 20. vendor/laravel/framework/src/Illuminate/View/Engines/PhpEngine.php:58

select * from `twill_faq_slugs` where `twill_faq_slugs`.`twill_faq_id` in (4) and `twill_faq_slugs`.`deleted_at` is null

4.78mstwenty4_siteFaqItem.php#15

Backtrace

20. app/View/Components/FaqItem.php:15
21. vendor/laravel/framework/src/Illuminate/View/Component.php:102
22. view::site.blocks.faq:58
24. vendor/laravel/framework/src/Illuminate/Filesystem/Filesystem.php:110
25. vendor/laravel/framework/src/Illuminate/View/Engines/PhpEngine.php:58

App\Models\User\Tutor: 6Tutor.php
App\Models\HomeworkLibrary\HomeworkLibrary: 6HomeworkLibrary.php
App\Models\Twill\Slugs\TwillFaqSlug: 6TwillFaqSlug.php
App\Models\Twill\TwillFaq: 4TwillFaq.php
App\Models\Subject: 2Subject.php
App\Models\SubjectCat: 2SubjectCat.php
App\Models\Meta: 1Meta.php

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/subjects/Mathematics/Number-Theory

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