Gọi d là ước chung nguyên tố của 2n + 7 và 5n + 2
\(\Rightarrow\left\{\begin{matrix}2n+7⋮d\\5n+2⋮d\end{matrix}\right.\)
+) Vì : 2n + 7 \(⋮\) d ; 5 \(\in N\Rightarrow5\left(2n+7\right)⋮d\Rightarrow10n+35⋮d\)
+) Vì : 5n + 2 \(⋮d;2\in N\Rightarrow2\left(5n+2\right)⋮d\Rightarrow10n+4⋮d\)
Mà : \(10n+35⋮d\)
\(\Rightarrow\left(10n+35\right)-\left(10n+4\right)⋮d\)
\(\Rightarrow10n+35-10n-4⋮d\)
\(\Rightarrow31⋮d\Rightarrow d\in\left\{-1;1;-31;31\right\}\)
Mà d nguyên tố \(\Rightarrow d=31\)
Với d = 31
\(\Rightarrow5n+2⋮31\) ; \(6\in N\) \(\Rightarrow6\left(5n+2\right)⋮31\Rightarrow30n+12⋮31\)
\(\Rightarrow31n-n+12⋮31\Rightarrow31n-\left(n-12\right)⋮31\)
\(\Rightarrow n-12⋮31\Rightarrow n-12=31k\Rightarrow n=31k+12\)
Với n = 31k + 12 \(\left(k\in N\right)\)
2n + 7 = 2 ( 31k + 12 ) + 7 = 62k + 24 + 7 = 62k + 31
= 31 ( 2k + 1 ) \(⋮\) 31
5n + 2 = 5 ( 31k + 12 ) + 2 = 105k + 60 + 2 = 105k + 62
= 31 ( 5k + 2 ) \(⋮\) 31
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