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","type":"extended-text","version":1,"note":"","noteHtml":"","textTranslaterHtml":"","typeTextTranslater":null}],"direction":null,"format":"","indent":0,"type":"paragraph","version":1,"textFormat":0,"textStyle":""},{"children":[{"detail":0,"format":0,"mode":"normal","style":"color: rgb(0, 0, 0)","text":"- Đưa về hai lũy thừa cùng số mũ để giải bài toán tìm cơ số: $x^n = y^n \\Rightarrow x = y$.","type":"extended-text","version":1,"note":"","noteHtml":"","textTranslaterHtml":"","typeTextTranslater":null}],"direction":null,"format":"","indent":0,"type":"paragraph","version":1,"textFormat":0,"textStyle":""},{"children":[{"detail":0,"format":0,"mode":"normal","style":"color: rgb(0, 0, 0)","text":"- Nếu cho khoảng, dùng phương pháp đánh giá: $a^m < a^x < a^n \\Rightarrow m < x < n$.","type":"extended-text","version":1,"note":"","noteHtml":"","textTranslaterHtml":"","typeTextTranslater":null}],"direction":null,"format":"","indent":0,"type":"paragraph","version":1,"textFormat":0,"textStyle":""}],"direction":null,"format":"","indent":0,"type":"quote","version":1},{"children":[{"detail":0,"format":1,"mode":"normal","style":"","text":"Ví dụ 2. ","type":"extended-text","version":1,"note":"","noteHtml":"","textTranslaterHtml":"","typeTextTranslater":null},{"detail":0,"format":0,"mode":"normal","style":"","text":"Tìm số tự nhiên $n$ biết rằng: $25 < 5^n < 3125$.","type":"extended-text","version":1,"note":"","noteHtml":"","textTranslaterHtml":"","typeTextTranslater":null}],"direction":null,"format":"justify","indent":0,"type":"paragraph","version":1,"textFormat":0,"textStyle":""},{"children":[{"detail":0,"format":2,"mode":"normal","style":"","text":"Lời giải","type":"extended-text","version":1,"note":"","noteHtml":"","textTranslaterHtml":"","typeTextTranslater":null}],"direction":null,"format":"justify","indent":0,"type":"paragraph","version":1,"textFormat":0,"textStyle":""},{"children":[{"detail":0,"format":0,"mode":"normal","style":"","text":"Ta đưa các số về cùng lũy thừa cơ số 5: $25 = 5^2$ và $3125 = 5^5$.","type":"extended-text","version":1,"note":"","noteHtml":"","textTranslaterHtml":"","typeTextTranslater":null}],"direction":null,"format":"justify","indent":0,"type":"paragraph","version":1,"textFormat":0,"textStyle":""},{"children":[{"detail":0,"format":0,"mode":"normal","style":"","text":"Bất phương trình trở thành: $5^2 < 5^n < 5^5$.","type":"extended-text","version":1,"note":"","noteHtml":"","textTranslaterHtml":"","typeTextTranslater":null}],"direction":null,"format":"justify","indent":0,"type":"paragraph","version":1,"textFormat":0,"textStyle":""},{"children":[{"detail":0,"format":0,"mode":"normal","style":"","text":"Do cơ số $5 > 1$ nên ta suy ra: $2 < n < 5$.","type":"extended-text","version":1,"note":"","noteHtml":"","textTranslaterHtml":"","typeTextTranslater":null}],"direction":null,"format":"justify","indent":0,"type":"paragraph","version":1,"textFormat":0,"textStyle":""},{"children":[{"detail":0,"format":0,"mode":"normal","style":"","text":"Vì $n$ là số tự nhiên nên $n$ nhận các giá trị $3$ hoặc $4$.","type":"extended-text","version":1,"note":"","noteHtml":"","textTranslaterHtml":"","typeTextTranslater":null}],"direction":null,"format":"justify","indent":0,"type":"paragraph","version":1,"textFormat":0,"textStyle":""},{"children":[{"children":[{"detail":0,"format":0,"mode":"normal","style":"","text":"Câu hỏi:","type":"extended-text","version":1,"note":"","noteHtml":"","textTranslaterHtml":"","typeTextTranslater":null}],"direction":null,"format":"","indent":0,"type":"heading","version":1,"tag":"h3"},{"children":[{"children":[{"detail":0,"format":0,"mode":"normal","style":"","text":"@209326533178@@209326523521@","type":"extended-text","version":1,"note":"","noteHtml":"","textTranslaterHtml":"","typeTextTranslater":null}],"direction":null,"format":"","indent":0,"type":"paragraph","version":1,"textFormat":0,"textStyle":""}],"direction":null,"format":"","indent":0,"type":"olm-special-child","version":1}],"direction":null,"format":"","indent":0,"type":"olm-special-parent","version":1,"name":"question"},{"children":[{"detail":0,"format":1,"mode":"normal","style":"","text":"Dạng 3. Bài toán tìm chữ số tận cùng","type":"extended-text","version":1,"note":"","noteHtml":"","textTranslaterHtml":"","typeTextTranslater":null}],"direction":null,"format":"justify","indent":0,"type":"heading","version":1,"tag":"h3"},{"children":[{"children":[{"detail":0,"format":1,"mode":"normal","style":"color: rgb(0, 0, 0)","text":"Phương pháp giải","type":"extended-text","version":1,"note":"","noteHtml":"","textTranslaterHtml":"","typeTextTranslater":null}],"direction":null,"format":"","indent":0,"type":"paragraph","version":1,"textFormat":0,"textStyle":""},{"children":[{"detail":0,"format":0,"mode":"normal","style":"color: rgb(0, 0, 0)","text":"Dựa vào số lượng chữ số tận cùng cần tìm để xét chu kì của số mũ (chu kì $4$ đối với 1 chữ số, chu kì $20$ đối với 2 chữ số, chu kì $100$ đối với 3 chữ số). ","type":"extended-text","version":1,"note":"","noteHtml":"","textTranslaterHtml":"","typeTextTranslater":null}],"direction":null,"format":"","indent":0,"type":"paragraph","version":1,"textFormat":0,"textStyle":""},{"children":[{"detail":0,"format":0,"mode":"normal","style":"color: rgb(0, 0, 0)","text":"Tách số mũ để làm xuất hiện các giá trị lũy thừa có tận cùng rơi vào trường hợp đặc biệt ($...1;\\, ...6;\\, ...25;\\, ...76$).","type":"extended-text","version":1,"note":"","noteHtml":"","textTranslaterHtml":"","typeTextTranslater":null}],"direction":null,"format":"","indent":0,"type":"paragraph","version":1,"textFormat":0,"textStyle":""}],"direction":null,"format":"","indent":0,"type":"quote","version":1},{"children":[{"detail":0,"format":1,"mode":"normal","style":"","text":"Ví dụ 3","type":"extended-text","version":1,"note":"","noteHtml":"","textTranslaterHtml":"","typeTextTranslater":null},{"detail":0,"format":0,"mode":"normal","style":"","text":". Tìm chữ số tận cùng của $7^{2\\,008}$ và tìm hai chữ số tận cùng của $2^{200}$.","type":"extended-text","version":1,"note":"","noteHtml":"","textTranslaterHtml":"","typeTextTranslater":null}],"direction":null,"format":"justify","indent":0,"type":"paragraph","version":1,"textFormat":0,"textStyle":""},{"children":[{"detail":0,"format":2,"mode":"normal","style":"","text":"Lời giải ","type":"extended-text","version":1,"note":"","noteHtml":"","textTranslaterHtml":"","typeTextTranslater":null}],"direction":null,"format":"justify","indent":0,"type":"paragraph","version":1,"textFormat":0,"textStyle":""},{"children":[{"detail":0,"format":0,"mode":"normal","style":"","text":"+ ","type":"extended-text","version":1,"note":"","noteHtml":"","textTranslaterHtml":"","typeTextTranslater":null},{"detail":0,"format":1,"mode":"normal","style":"","text":"Tìm 1 chữ số tận cùng:","type":"extended-text","version":1,"note":"","noteHtml":"","textTranslaterHtml":"","typeTextTranslater":null},{"detail":0,"format":0,"mode":"normal","style":"","text":" Ta biết các số tận cùng là $7$ nâng lên lũy thừa bậc $4n$ luôn tận cùng là $1$. ","type":"extended-text","version":1,"note":"","noteHtml":"","textTranslaterHtml":"","typeTextTranslater":null}],"direction":null,"format":"justify","indent":0,"type":"paragraph","version":1,"textFormat":0,"textStyle":""},{"children":[{"detail":0,"format":0,"mode":"normal","style":"","text":"Phân tích: $7^{2\\,008} = (7^4)^{502}$. ","type":"extended-text","version":1,"note":"","noteHtml":"","textTranslaterHtml":"","typeTextTranslater":null}],"direction":null,"format":"justify","indent":0,"type":"paragraph","version":1,"textFormat":0,"textStyle":""},{"children":[{"detail":0,"format":0,"mode":"normal","style":"","text":"Cơ số bên trong ngoặc có tận cùng là $1$, nên khi nâng lên bậc $502$ vẫn giữ nguyên tận cùng là $1$. ","type":"extended-text","version":1,"note":"","noteHtml":"","textTranslaterHtml":"","typeTextTranslater":null}],"direction":null,"format":"justify","indent":0,"type":"paragraph","version":1,"textFormat":0,"textStyle":""},{"children":[{"detail":0,"format":0,"mode":"normal","style":"","text":"Vậy $7^{2\\,008}$ tận cùng bằng $1$.","type":"extended-text","version":1,"note":"","noteHtml":"","textTranslaterHtml":"","typeTextTranslater":null}],"direction":null,"format":"justify","indent":0,"type":"paragraph","version":1,"textFormat":0,"textStyle":""},{"children":[{"detail":0,"format":0,"mode":"normal","style":"","text":"+ ","type":"extended-text","version":1,"note":"","noteHtml":"","textTranslaterHtml":"","typeTextTranslater":null},{"detail":0,"format":1,"mode":"normal","style":"","text":"Tìm 2 chữ số tận cùng:","type":"extended-text","version":1,"note":"","noteHtml":"","textTranslaterHtml":"","typeTextTranslater":null},{"detail":0,"format":0,"mode":"normal","style":"","text":" Ta biết số chẵn mũ chu kì $20$ (cụ thể $a^{20k}$ với a chẵn khác tận cùng $0$) luôn tận cùng là $76$. ","type":"extended-text","version":1,"note":"","noteHtml":"","textTranslaterHtml":"","typeTextTranslater":null}],"direction":null,"format":"justify","indent":0,"type":"paragraph","version":1,"textFormat":0,"textStyle":""},{"children":[{"detail":0,"format":0,"mode":"normal","style":"","text":"Phân tích $2^{200} = (2^{20})^{10} = 76^{10}$. Mà các số có hai chữ số tận cùng là $76$ nâng lên lũy thừa bất kì khác $0$ luôn giữ nguyên tận cùng là $76$. ","type":"extended-text","version":1,"note":"","noteHtml":"","textTranslaterHtml":"","typeTextTranslater":null}],"direction":null,"format":"justify","indent":0,"type":"paragraph","version":1,"textFormat":0,"textStyle":""},{"children":[{"detail":0,"format":0,"mode":"normal","style":"","text":"Vậy $2^{200}$ có hai chữ số tận cùng là $76$.","type":"extended-text","version":1,"note":"","noteHtml":"","textTranslaterHtml":"","typeTextTranslater":null}],"direction":null,"format":"justify","indent":0,"type":"paragraph","version":1,"textFormat":0,"textStyle":""},{"children":[{"children":[{"detail":0,"format":0,"mode":"normal","style":"","text":"Câu hỏi:","type":"extended-text","version":1,"note":"","noteHtml":"","textTranslaterHtml":"","typeTextTranslater":null}],"direction":null,"format":"","indent":0,"type":"heading","version":1,"tag":"h3"},{"children":[{"children":[{"detail":0,"format":0,"mode":"normal","style":"","text":"@209383167187@@209383177486@","type":"extended-text","version":1,"note":"","noteHtml":"","textTranslaterHtml":"","typeTextTranslater":null}],"direction":null,"format":"","indent":0,"type":"paragraph","version":1,"textFormat":0,"textStyle":""}],"direction":null,"format":"","indent":0,"type":"olm-special-child","version":1}],"direction":null,"format":"","indent":0,"type":"olm-special-parent","version":1,"name":"question"}],"direction":null,"format":"","indent":0,"type":"root","version":1}}

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