a) \(\overline{aaa}=\overline{a00}+\overline{a0}+a=a.100+a.10+a.1\\ =a.\left(100+10+1\right)=a.111=a.37.3⋮3\) (dpcm)

b) \(\overline{ab}+\overline{ba}=\overline{a0}+b+\overline{b0}+a\\ =a.10+b+b.10+a\\ =a.\left(10+1\right)+b.\left(1+10\right)\\ =a.11+b.11\\ =11\left(a+b\right)⋮11\) (dpcm)

c) \(\overline{ab}-\overline{ba}=\overline{a0}+b-\left(\overline{b0}+a\right)\\ =a.10+b-b.10-a\\ =a.\left(10-1\right)+b.\left(1-10\right)\\ =a.9+b.\left(-9\right)\\ =9.\left(a-b\right)⋮9\) (dpcm)

d) \(\overline{abcabc}=\overline{abc000}+\overline{abc}\\ =\overline{abc}.1000+\overline{abc}.1\\ =\overline{abc}.1001=\overline{abc}.11.91⋮11\) (dpcm)

\(\dfrac{1}{4}\) m hay dm hay như nào em ơi?

Chiều cao hình bình hành là:

\(\dfrac{5}{6}:\dfrac{1}{4}=\dfrac{10}{3}\left(m\right)\)

Đáp số : \(\dfrac{10}{3}m\)

Công thức:

\(h=S:a\)

\(250:\left\{5\cdot\left[1997-1869\right]-78\right\}\)

\(=250:\left\{5\cdot128-78\right\}\)

\(=250:\left\{640-78\right\}=\dfrac{250}{562}=\dfrac{125}{281}\)

\(6\cdot1235\cdot20-5\cdot235\cdot24\)

\(=120\cdot1235-120\cdot235=120\cdot1000\)

=120000

176.92

176.92

\(\dfrac{9}{4}-\dfrac{1}{4}:y-\dfrac{1}{2}=\dfrac{2}{3}\\ =>\dfrac{1}{4}:y=\dfrac{9}{4}-\dfrac{1}{2}-\dfrac{2}{3}\\ =>\dfrac{1}{4}:y=\dfrac{27}{12}-\dfrac{6}{12}-\dfrac{8}{12}\\ =>\dfrac{1}{4}:y=\dfrac{13}{12}\\ =>y=\dfrac{1}{4}:\dfrac{13}{12}\\ =>y=\dfrac{3}{13}\)

`9/4- 1/4:y - 1/2 = 2/3`

`<=> 9/4 - 1/4:y - 2/4 = 2/3`

`<=> 7/4 - 1/4:y= 2/3`

`<=> 1/4:y = 7/4 - 2/3`

`<=> 1/4:y = 21/12 - 8/12`

`<=> 1/4:y = 13/12`

`<=> y = 1/4 : 13/12`

`<=> y = 1/4 . 12/13`

`<=> y = 3/13`

Vậy ..

\(5,1\times y-y:10=36,25\times4\)

\(5,1\times y-y\times0,1=145\)

\(\left(5,1-0,1\right)\times y=145\)

\(5\times y=145\)

\(y=145:5\)

\(y=29\)

`5,1 xx y - y : 10 = 36,25 xx 4`

`=>5,1 xx y - y xx 0,1 = 36 x 4 + 0,25 x 4`

`=>y xx (5,1 - 0,1) = 144+1`

`=>y xx 5 = 145`

`=>y=145:5`

`=>y=29`

Vậy: ... 

`x(x-5)=0`

TH1: `x=0`

TH2: `x-5=0`

`=>x=0+5`

`=>x=5`

Vậy: `x∈{0;5}`

Bổ sung cho @HT.Phong

Kết luận: \(x\) \(\in\) {0; 5}

\(124\times\left(56+67\right)-24\times\left(67+56\right)\\ =124\times123-24\times123\\ =\left(124-24\right)\times123\\ =100\times123\\ =12300\)

  124 x (56 + 67) - 24 x (67 + 56)

= (56 + 67) x (125 - 24)

= 123 x 100 

= 12300

\(\dfrac{2}{3}\times\dfrac{8}{17}+\dfrac{2}{3}\times\dfrac{5}{17}+\dfrac{2}{3}\times\dfrac{4}{17}\)

\(=\dfrac{2}{3}\times\left(\dfrac{8}{17}+\dfrac{5}{17}+\dfrac{4}{17}\right)\)

\(=\dfrac{2}{3}\times1\)

\(=\dfrac{2}{3}\)