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Tüm problemler modüler aritmetik ve Euler Teoremi kullanılarak adım adım çözülmektedir."]},"endTime":1314,"title":"Euler Teoremi ve Uygulamaları Dersi","beginTime":0}],"fullResult":[{"index":0,"title":"Euler Teoremi ve Uygulamaları","list":{"type":"unordered","items":["Euler teoremi, modüler aritmetikte a ile n aralarında asal (EBOB(a,n)=1) olmak üzere a^φ(n) ≡ 1 (mod n) olduğunu belirtir.","φ fonksiyonu, n'den küçük eşit olan ve n ile aralarında asal olan pozitif tam sayıların sayısını verir.","Euler teoremi, Fermat ve Wilson teoremlerinin ispatları da önceden ele alınmış durumdadır."]},"beginTime":2,"endTime":48,"href":"/video/preview/10739128117737828713?parent-reqid=1769609638754757-16772600860499602845-balancer-l7leveler-kubr-yp-klg-18-BAL&text=Euler13&t=2&ask_summarization=1"},{"index":1,"title":"Euler Teoremi ile Problem Çözümü","list":{"type":"unordered","items":["7^2018 sayısının 45 ile bölümünden kalanı bulmak için öncelikle φ(45) hesaplanır ve 24 olarak bulunur.","Euler teoremi gereği 7^24 ≡ 1 (mod 45) olduğu için, 2018 sayısının 24'e bölümünden kalan 2 olduğu görülmektedir.","Sonuç olarak 7^2018 ≡ 7^2 ≡ 49 ≡ 4 (mod 45) olduğu bulunur ve kalan 4 olarak hesaplanır."]},"beginTime":48,"endTime":202,"href":"/video/preview/10739128117737828713?parent-reqid=1769609638754757-16772600860499602845-balancer-l7leveler-kubr-yp-klg-18-BAL&text=Euler13&t=48&ask_summarization=1"},{"index":2,"title":"Alternatif Çözüm Yöntemi","list":{"type":"unordered","items":["Euler teoremi, Fermat teoreminin genelleştirilmiş bir biçimidir ancak bazen Fermat teoremi ve Çin Kalan Teoremi birleştirilerek daha etkili sonuçlar elde edilebilir.","Çin Kalan Teoremi ile 45'in çarpanları olan 5 ve 9 modüllerinde hesaplamalar yapılır ve sonuçlar birleştirilerek çözüm bulunur.","Bu yöntemle 7^12 ≡ 1 (mod 45) elde edilir ve 2018 sayısının 12'ye bölümünden kalan 2 olduğu için tekrar kalan 4 olarak bulunur."]},"beginTime":202,"endTime":486,"href":"/video/preview/10739128117737828713?parent-reqid=1769609638754757-16772600860499602845-balancer-l7leveler-kubr-yp-klg-18-BAL&text=Euler13&t=202&ask_summarization=1"},{"index":3,"title":"Euler Teoreminin Zorunlu Kullanım Alanları","list":{"type":"unordered","items":["13^1502 sayısının 125 ile bölümünden kalanı bulmak için, 125'in sadece 5'lerden oluştuğu (5^3) için alternatif yöntemler uygulanamaz.","Euler teoremi gereği φ(125) = 100 olarak hesaplanır ve 13^100 ≡ 1 (mod 125) elde edilir.","1502 sayısının 100'e bölümünden kalan 2 olduğu için 13^1502 ≡ 13^2 ≡ 169 ≡ 44 (mod 125) olarak hesaplanır."]},"beginTime":486,"endTime":645,"href":"/video/preview/10739128117737828713?parent-reqid=1769609638754757-16772600860499602845-balancer-l7leveler-kubr-yp-klg-18-BAL&text=Euler13&t=486&ask_summarization=1"},{"index":4,"title":"Modüler Aritmetik Problemi Çözümü","list":{"type":"unordered","items":["275 üzeri 2001 sayısının 54 ile bölümünden kalanı hesaplanıyor.","54 sayısı 3'ün küpü çarpı 2 olarak çarpanlara ayrılır ve Euler teoremi veya Çin Kalan Teoremi kullanılabilir.","Tabanı küçültmek için 275 sayısının 54'e bölünmesi 5 kalanını verir, bu nedenle 5 üzeri 2001 sayısının mod 54'teki değerine odaklanılır."]},"beginTime":651,"endTime":732,"href":"/video/preview/10739128117737828713?parent-reqid=1769609638754757-16772600860499602845-balancer-l7leveler-kubr-yp-klg-18-BAL&text=Euler13&t=651&ask_summarization=1"},{"index":5,"title":"Euler Teoremi Uygulaması","list":{"type":"unordered","items":["5 sayısı ile 54 aralarında asal olduğundan Euler teoremi uygulanabilir.","φ(54) = 18 olarak hesaplanır ve Euler teoremi gereği 5 üzeri 18 sayısı mod 54'te 1'e denktir.","2001 sayısının 18 ile bölümünden kalan 3 olduğundan, 5 üzeri 2001 sayısı 5 üzeri 3'e denktir."]},"beginTime":732,"endTime":874,"href":"/video/preview/10739128117737828713?parent-reqid=1769609638754757-16772600860499602845-balancer-l7leveler-kubr-yp-klg-18-BAL&text=Euler13&t=732&ask_summarization=1"},{"index":6,"title":"Çin Kalan Teoremi Uygulaması","list":{"type":"unordered","items":["Alternatif olarak Çin Kalan Teoremi uygulanabilir.","5 üzeri 2001 sayısı mod 2'de 1'e denktir ve mod 27'de 17'e denktir.","Bu denklemlerden x denk 17 mod 54 olarak bulunur."]},"beginTime":874,"endTime":1062,"href":"/video/preview/10739128117737828713?parent-reqid=1769609638754757-16772600860499602845-balancer-l7leveler-kubr-yp-klg-18-BAL&text=Euler13&t=874&ask_summarization=1"},{"index":7,"title":"Karınca Problemi","list":{"type":"unordered","items":["101 odalı bir yuvasında karıncalar ilk odaya 1 küp şeker, ikinci odaya 2 küp şeker, üçüncü odaya 4 küp şeker koyarak 101. odaya kadar 2 üzeri 100 kadar şeker koyarlar.","Toplam şeker sayısı 2 üzeri 101 eksi 1 olarak hesaplanır.","Toplam 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Konuşmacı, üçgenlerde önemli geometrik özellikler ve noktaları göstermektedir.","Video, eşkenar olmayan bir üçgenin kenar ortayları, medyanları ve yüksekliklerini çizerek başlıyor. Bu üçgenin üç önemli noktasının (kenar ortayların kesiştiği nokta, medyanların kesiştiği ağırlık merkezi ve yüksekliklerin kesiştiği yükseklik merkezi) aynı doğru üzerinde yer aldığı gösteriliyor. Bu doğru Euler doğrusu olarak adlandırılıyor. Konuşmacı, Euler'in bu keşfi ve diğer matematiksel katkılarını kısaca anlatarak, Euler doğrusunun üzerinde bulunan dokuz nokta çemberinin merkezi gibi ilginç noktaları da gösteriyor."]},"endTime":319,"title":"Üçgenlerde Euler Doğrusu ve Özel Noktalar","beginTime":0}],"fullResult":[{"index":0,"title":"Üçgenin Özel Noktaları","list":{"type":"unordered","items":["Eşkenar olmayan bir üçgenin kenar ortayları, medyanları ve yükseklikleri çiziliyor.","Kenar ortayları, medyanlar ve yüksekliklerin kesişim noktaları sırasıyla çevrel merkez, ağırlık merkezi ve yükseklik merkezi olarak adlandırılıyor.","Bu üç noktanın şaşırtıcı şekilde aynı doğru üzerinde yer aldığı belirtiliyor."]},"beginTime":1,"endTime":180,"href":"/video/preview/1986425349052575356?parent-reqid=1769609638754757-16772600860499602845-balancer-l7leveler-kubr-yp-klg-18-BAL&text=Euler13&t=1&ask_summarization=1"},{"index":1,"title":"Euler Doğrusu","list":{"type":"unordered","items":["Bu özel doğrunun ismi Euler doğrusu olarak 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kaynak kodlarım için githubımı ziyaret edebilirsiniz: https://github.com/Daymenion Merhaba arkadaşlar, ben Mehmet. Youtube'da neler yapacağımı kısaca anlatayım, sonrasında da bu...","preview":{"posterSrc":"//avatars.mds.yandex.net/get-vthumb/1784436/eb0a8ff634a7ce184195a61ebb9918fe/564x318_1","videoSrc":"https://video-preview.s3.yandex.net/BYxTGgEAAAA.mp4","videoType":"video/mp4"},"target":"_self","position":"7","reqid":"1769609638754757-16772600860499602845-balancer-l7leveler-kubr-yp-klg-18-BAL","summary":{"fullTextUrl":"/int_search_summary?data=http%3A%2F%2Fwww.youtube.com%2Fwatch%3Fv%3DDIifCFXncfY","linkTemplate":"/video/preview/4058321426327301994?parent-reqid=1769609638754757-16772600860499602845-balancer-l7leveler-kubr-yp-klg-18-BAL&text=Euler13&t=%%timestamp%%&ask_summarization=1"},"isAdultDoc":false,"relatedParams":{"text":"Project Euler Problem 13 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