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請教一題倍數問題

請教一題倍數問題

感謝老師的指導

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2022-11-21 13:30

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淺見提供,有錯還望不吝指教

若\(\displaystyle x=7p+1 \ or \ 7p+6 \Rightarrow x^4\equiv 1\ (mod\ 7)\)
若\(\displaystyle x=7p+2 \ or \ 7p+5 \Rightarrow x^4\equiv 2\ (mod\ 7)\)
若\(\displaystyle x=7p+3 \ or \ 7p+4 \Rightarrow x^4\equiv 4\ (mod\ 7)\)
其中\(\displaystyle p\in \mathbb{N} \cup \{0 \}\)
並且
若\(\displaystyle x=3k+1  \Rightarrow 4^x\equiv 4 \ (mod\ 7)\)
若\(\displaystyle x=3k+2  \Rightarrow 4^x\equiv 2 \ (mod\ 7)\)
若\(\displaystyle x=3k  \Rightarrow 4^x\equiv 1 \ (mod\ 7)\)
其中\(\displaystyle k \in \mathbb{N} \cup \{0 \}\)

接下就是分類
\(\displaystyle x=3k=7p+1 \Rightarrow x=15+21N\)
\(\displaystyle x=3k=7p+6 \Rightarrow x=6+21N\)
\(\displaystyle x=3k+1=7p+3 \Rightarrow x=10+21N\)
\(\displaystyle x=3k+1=7p+4 \Rightarrow x=4+21N\)
\(\displaystyle x=3k+2=7p+2 \Rightarrow x=2+21N\)
\(\displaystyle x=3k+2=7p+5 \Rightarrow x=5+21N\),以上個數在範圍之下皆為10個,故共有60個數滿足題意

PS.  \(x\)為7的倍數必無法滿足題意,故不考慮

[ 本帖最後由 satsuki931000 於 2022-11-21 14:32 編輯 ]

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