Tìm n để : a, 13 -2n chia hết cho 3n+1 b, n^2-n+1/n-2 là số nguyên c, 5n2 -3n+2 chia hết cho n-2

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1. Giải thích các bước giải:

   a.Ta có:

   $13-2n\quad\vdots\quad 3n+1$

   $\to 3(13-2n)\quad\vdots\quad 3n+1$

   $\to 39-6n\quad\vdots\quad 3n+1$

   $\to 39+2-6n-2\quad\vdots\quad 3n+1$ 

   $\to 41-2(3n+1)\quad\vdots\quad 3n+1$ 

   $\to 41\quad\vdots\quad 3n+1$ 

   $\to  3n+1\in U(41)$ vì $n\in Z$

   Mà $3n+1$ chia $3$ dư $1$

   $\to 3n+1\in\{-41,1\}$

   $\to 3n\in\{-42, 0\}$

   $\to n\in\{-14, 0\}$

   b.Để $\dfrac{n^2-n+1}{n-2}$ là số nguyên

   $\to n^2-n+1\quad\vdots\quad n-2$ vì $n\in Z$

   $\to (n^2-2n)+(n-2)+3\quad\vdots\quad n-2$ 

   $\to n(n-2)+(n-2)+3\quad\vdots\quad n-2$ 

   $\to 3\quad\vdots\quad n-2$ 

   $\to n-2\in U(3)$ vì $n\in Z$

   $\to n-2\in\{1,3,-1,-3\}$

   $\to n\in\{3,5,1,-1\}$

   c.Ta có:

   $5n^2-3n+2\quad\vdots\quad n-2$

   $\to (5n^2-10n)+(7n-14)+16\quad\vdots\quad n-2$

   $\to 5n(n-2)+7(n-2)+16\quad\vdots\quad n-2$

   $\to 16\quad\vdots\quad n-2$

   $\to n-2\in U(16)$ vì $n\in Z$

   $\to n-2\in\{1,2,4,16,-1,-2,-4,-16\}$

   $\to n\in\{3,4,6,18,1,0,-2,-14\}$

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2. a.Ta có

   13−2n⋮3n+113−2n⋮3n+1

   =>3(13−2n)⋮3n+1→3(13−2n)⋮3n+1

   =>39−6n⋮3n+1→39−6n⋮3n+1

   =>39+2−6n−2⋮3n+1→39+2−6n−2⋮3n+1 

   =>41−2(3n+1)⋮3n+1→41−2(3n+1)⋮3n+1 

   =>41⋮3n+1→41⋮3n+1 

   =>3n+1∈U(41)→3n+1∈U(41) vì n∈Zn∈Z

   Mà 3n+13n+1 chia 33 dư 11

   =>3n+1∈{−41,1}→3n+1∈{−41,1}

   =>3n∈{−42,0}→3n∈{−42,0}

   =>n∈{−14,0}→n∈{−14,0}

   b. Để n2−n+1n−2n2−n+1n−2 là số nguyên

   Thì: n2−n+1⋮n−2→n2−n+1⋮n−2 vì n∈Zn∈Z

   =>(n2−2n)+(n−2)+3⋮n−2→(n2−2n)+(n−2)+3⋮n−2 

   =>n(n−2)+(n−2)+3⋮n−2→n(n−2)+(n−2)+3⋮n−2 

   =>3⋮n−2→3⋮n−2 

   =>n−2∈U(3)→n−2∈U(3) vì n∈Zn∈Z

   =>n−2∈{1,3,−1,−3}→n−2∈{1,3,−1,−3}

   =>n∈{3,5,1,−1}→n∈{3,5,1,−1}

   c.Ta có:

   5n2−3n+2⋮n−25n2−3n+2⋮n−2

   =>(5n2−10n)+(7n−14)+16⋮n−2→(5n2−10n)+(7n−14)+16⋮n−2

   =>5n(n−2)+7(n−2)+16⋮n−2→5n(n−2)+7(n−2)+16⋮n−2→16⋮n−2→16⋮n−2

   =>n−2∈U(16)→n−2∈U(16) vì n∈Zn∈Z

   =>n−2∈{1,2,4,16,−1,−2,−4,−16}→n−2∈{1,2,4,16,−1,−2,−4,−16}

   =>n∈{3,4,6,18,1,0,−2,−14}

    

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