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Ta cần chứng minh rằng:

\(6^{7260} \&\text{nbsp};\text{chia}\&\text{nbsp};\text{h} \overset{ˊ}{\hat{\text{e}}} \text{t}\&\text{nbsp};\text{cho}\&\text{nbsp};\text{c}ả\&\text{nbsp}; 7 \&\text{nbsp};\text{v} \overset{ˋ}{\text{a}} \&\text{nbsp}; 43.\)

🧠 Ý tưởng giải:

Chứng minh \(6^{7260} \equiv 0 \left(\right. m o d 7 \left.\right)\) và \(6^{7260} \equiv 0 \left(\right. m o d 43 \left.\right)\) là sai → vì rõ ràng \(6 < 7\), \(6 < 43\), nên không thể chia hết.

Nhưng có vẻ bạn đang muốn chứng minh:

\(6^{7260} \equiv 1 \left(\right. m o d 7 \left.\right) \text{v} \overset{ˋ}{\text{a}} 6^{7260} \equiv 1 \left(\right. m o d 43 \left.\right)\)

hoặc:

\(6^{7260} \equiv 1 \left(\right. m o d 301 \left.\right)\)

Vì 7 và 43 là các số nguyên tố, và:

\(7 \times 43 = 301\)

✅ Vậy ta sẽ chứng minh:

\(6^{7260} \equiv 1 \left(\right. m o d 7 \left.\right) \text{v} \overset{ˋ}{\text{a}} 6^{7260} \equiv 1 \left(\right. m o d 43 \left.\right)\)

✳️ Bước 1: Chứng minh \(6^{7260} \equiv 1 \left(\right. m o d 7 \left.\right)\)

Sử dụng Định lý Fermat nhỏ:

Với \(p\) là số nguyên tố và \(a\) không chia hết cho \(p\), thì:

\(a^{p - 1} \equiv 1 \left(\right. m o d p \left.\right)\)

• Với \(a = 6\), \(p = 7\), ta có:

\(6^{6} \equiv 1 \left(\right. m o d 7 \left.\right)\)

→ Mà \(7260\) chia hết cho \(6\), vì:

\(7260 \div 6 = 1210\)

⇒

\(6^{7260} = \left(\right. 6^{6} \left.\right)^{1210} \equiv 1^{1210} \equiv 1 \left(\right. m o d 7 \left.\right)\)

✳️ Bước 2: Chứng minh \(6^{7260} \equiv 1 \left(\right. m o d 43 \left.\right)\)

• \(43\) là số nguyên tố, \(6\) không chia hết cho \(43\)

Áp dụng định lý Fermat:

\(6^{42} \equiv 1 \left(\right. m o d 43 \left.\right)\)

Vì:

\(7260 \div 42 = 172.857... \Rightarrow t a k i ể m t r a 7260 c \overset{ˊ}{o} c h i a h \overset{ˊ}{\hat{e}} t c h o 42 k h \hat{o} n g ?\)\(7260 \div 42 = 172.857... \rightarrow k h \hat{o} n g c h i a h \overset{ˊ}{\hat{e}} t !\)

Nhưng ta có thể viết:

\(7260 = 42 \times 172 + 36\)

⇒

\(6^{7260} = \left(\right. 6^{42} \left.\right)^{172} \cdot 6^{36} \equiv 1^{172} \cdot 6^{36} \equiv 6^{36} \left(\right. m o d 43 \left.\right)\)

Ta cần kiểm tra \(6^{36} m o d \textrm{ } \textrm{ } 43\). Đây khá lớn, nên thay vì tính trực tiếp, ta dùng chu kỳ modulo.

✳️ Tìm chu kỳ của \(6^{n} m o d \textrm{ } \textrm{ } 43\)

Tìm số nhỏ nhất \(k\) sao cho:

\(6^{k} \equiv 1 \left(\right. m o d 43 \left.\right)\)

Tức là tìm bậc của 6 modulo 43.

Ta thử dần:

• \(6^{1} = 6\)
• \(6^{2} = 36\)
• \(6^{3} = 216 m o d \textrm{ } \textrm{ } 43 = 216 - 5 \times 43 = 1 \Rightarrow 6^{3} \equiv 1 m o d \textrm{ } \textrm{ } 43\)

→ Vậy:

\(6^{3} \equiv 1 m o d \textrm{ } \textrm{ } 43 \Rightarrow \text{chu}\&\text{nbsp};\text{k} \overset{ˋ}{\text{y}} \&\text{nbsp};\text{c}ủ\text{a}\&\text{nbsp}; 6 m o d \textrm{ } \textrm{ } 43 = 3\)

Vì chu kỳ là 3, ta chia:

\(7260 \div 3 = 2420 \Rightarrow 6^{7260} = \left(\right. 6^{3} \left.\right)^{2420} \equiv 1^{2420} = 1 m o d \textrm{ } \textrm{ } 43\)

✅ Kết luận:

\(6^{7260} \equiv 1 m o d \textrm{ } \textrm{ } 7\)\(6^{7260} \equiv 1 m o d \textrm{ } \textrm{ } 43\)

⇒ Theo định lý Chinese Remainder Theorem, suy ra:

\(6^{7260} \equiv 1 m o d \textrm{ } \textrm{ } 301\)

✅ Trả lời:

6⁷²⁶⁰ chia cho cả 7 và 43 đều dư 1 ⇒ không chia hết, nhưng:

\(6^{7260} \equiv 1 m o d \textrm{ } \textrm{ } 7 \&\text{nbsp};\text{v} \overset{ˋ}{\text{a}} \&\text{nbsp}; m o d \textrm{ } \textrm{ } 43\)

Là chào

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I like country life. Because people in the countryside are more friendly than those who live in city.

In the countryside, the enviroment is very fresh and it have many trees. My mind is better: not busy, peaceful.

Many people say that: " the country life is very boring!" but I don't think so. I think although the countryside doesn't have many means of the entertainment but we can play traditionnal games.

In the city, people always want to work and work, they want to have much money. So, sometimes they forget many interesting things in their life.

But in the countryside, you can become another one : you are better, happier anh care of everybody. Here, people are a big family, if you are sad, they are sad with you. If you smile, they also smile with you.

Only thing I want to say " I love country life forever"