Question

[Hayato]

Tìm a,b,c biết ab.bc.=abbc

Ai nhanh mình tick

Answer 1

[T.Anh 2K7(siêu quậy)(тoáɴ๖ۣۜнọc)]

\(\overline{ab}.\overline{bc}=\overline{abbc}=100.\overline{ab}+\overline{bc}\left(1\right)\)

\(\Rightarrow100.\overline{ab}+\overline{bc}⋮\overline{ab\Rightarrow\overline{bc}⋮\overline{ab}\Rightarrow}\overline{bc}=k.\overline{ab}\left(k\inℕ^∗|k< 10\right)\left(2\right)\)

Ta có từ (1) suy ra

\(100.\overline{ab}+\overline{bc}⋮\overline{bc}\Rightarrow100.\overline{ab}+k.\overline{ab}⋮k.\overline{ab}\Rightarrow100+k⋮k\Rightarrow100⋮k\left(3\right)\)(Từ (2))

Mà 0< k<10(4)

Từ(3) và (4)

\(\Rightarrow k\in\left\{1;2;4;5\right\}\)

Với k=1,thay vào (2) suy ra\(\overline{ab}=\overline{bc}\)

\(\Rightarrow\overline{ab}.\overline{ab}=\overline{abab}\Rightarrow\overline{ab}^2=101.\overline{ab}\Rightarrow\overline{ab}=101\left(L\right)\)

Với k=2,thay vào (2) suy ra\(\overline{bc}=2.\overline{ab}\)

\(\Rightarrow2.\overline{ab}.\overline{ab}=100.\overline{ab}+2.\overline{ab}\Rightarrow2.\overline{ab}^2=102.\overline{ab}\Rightarrow\overline{ab}=51\Rightarrow\overline{bc}=102\left(L\right)\)

Với k=4,thay vào (2) suy ra\(\overline{bc}=4.\overline{ab}\)

\(\Rightarrow4.\overline{ab}.\overline{ab}=100.\overline{ab}+4.\overline{ab}\Rightarrow4.\overline{ab}^2=104.\overline{ab}\Rightarrow\overline{ab}=26\Rightarrow\overline{bc}=104\left(L\right)\)

Với k=5,thay vào (2) suy ra\(\overline{bc}=5.\overline{ab}\)

\(\Rightarrow5.\overline{ab}.\overline{ab}=100.\overline{ab}+5.\overline{ab}\Rightarrow5.\overline{ab}^2=105.\overline{ab}\Rightarrow\overline{ab}=21\Rightarrow\overline{bc}=105\left(L\right)\)

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