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TL:

\(\left(6a+11b\right)⋮31\)

\(\Rightarrow\left(6a+11b+31b\right)⋮31\)

\(\Rightarrow\left(6a+42b\right)⋮31\)

\(\Rightarrow6\left(a+7b\right)⋮31\)

\(\Rightarrow\left(a+7b\right)⋮31\)

\(\Rightarrow\)ĐPCM

~HT~

17+(−20)+23+(−26)+...+53+(−56)17+(−20)+23+(−26)+...+53+(−56) 

=[17+(−20)]+[23+(−26)]+...+[53+(−56)]=[17+(−20)]+[23+(−26)]+...+[53+(−56)]

 =(−3)+(−3)+...+(−3)=(−3)+(−3)+...+(−3)

=(−3).(7)=−21

=(17+39-156)+(-134-66)

=-100-200

= -300

= - 300 nha

* Chú ý đề thiếu : Tìm số tự nhiên n chứ bạn !

Theo bài ra ta có : \(\hept{\begin{cases}147-17⋮n\\193-11⋮n\end{cases}}\) \(\Rightarrow\)\(\hept{\begin{cases}130⋮n\\182⋮n\end{cases}}\) \(\Rightarrow\)\(n\inƯC\left(130,182\right)\)

130 = 2 . 5 . 13

182 = 2 . 7 . 13

\(\Rightarrow\)\(ƯCLN\left(130,182\right)=2.13=26\)

\(\Rightarrow\)\(ƯC\left(130,182\right)=Ư\left(26\right)=\left\{1;2;13;26\right\}\)

\(\Rightarrow\)\(n\in\left\{1;2;13;26\right\}\)

Vậy \(n\in\left\{1;2;13;26\right\}\)

[31.(x+3)].12=144

31.(x+3)       =144:12

31.(x+3)       = 12

x+3               = 12:31

x+3               = 12/31

x                   = 12/31 - 3

x                   = -81/31

  [31 . (x + 3)] . 12 = 144

   31 . (x + 3) = 144 : 12 

   31 . (x + 3) = 12

           x + 3 = \(\frac{12}{31}\)

                 x = \(\frac{12}{31}-3=\frac{12}{31}-\frac{93}{31}\)

                 x = \(\frac{-81}{31}\)

Vậy      x = \(\frac{-81}{31}\)