a) Do số dư là `36` nên số chia nhỏ nhất có thể là `37`

Số bị chia là: 

`37 . 15 + 36 = 591`

b) Ta có: `a : b = c (`dư `d)`

`=> a = bc + d (1) `

Theo bài ra ta có: `(a+15) : (b+5) = c (`dư `d)`

`=> a + 15 = c . (b + 5) + d`

`=> a + 15 = cb + 5c + d`

`=> a + 15 = a + 5c (`Từ `(1))`

`=> 5c = 15`

`=> c = 3`

Vậy thương là `3`

`(8x - 16)(x - 5) = 0`

`<=> 8x - 16 = 0` hoặc `x - 5 = 0`

`<=> 8x = 16` hoặc `x = 5`

`<=> x = 2` hoặc `x =  5`

Vậy `x = 2` hoặc `x = 5`

Tổng 5 số chẵn liên tiếp là

2k+(2k+2)+(2k+4)+(2k+6)+(2k+8)=10k+20=10(k+2)\(⋮10\)

Tổng 5 số lẻ liên tiếp là

(2k+1)+(2k+3)+(2k+5)+(2k+7)+(2k+9)=10k+25=10(k+2)+5 chia 10 dư 5

a: \(\dfrac{1}{5}x-\dfrac{2}{3}=\dfrac{5}{6}\)

=>\(\dfrac{1}{5}x=\dfrac{2}{3}+\dfrac{5}{6}=\dfrac{4}{6}+\dfrac{5}{6}=\dfrac{9}{6}=\dfrac{3}{2}\)

=>\(x=\dfrac{3}{2}:\dfrac{1}{5}=\dfrac{3}{2}\cdot5=\dfrac{15}{2}\)

b: \(\left(x+\dfrac{3}{4}\right)-\dfrac{5}{8}=\dfrac{3}{8}\)

=>\(x+\dfrac{3}{4}=\dfrac{3}{8}+\dfrac{5}{8}=1\)

=>\(x=1-\dfrac{3}{4}=\dfrac{1}{4}\)

c: \(x+\dfrac{8}{12}=\dfrac{3}{4}\)

=>\(x=\dfrac{3}{4}-\dfrac{8}{12}=\dfrac{9}{12}-\dfrac{8}{12}=\dfrac{1}{12}\)

d: \(x:\left[\dfrac{7}{2\cdot5}+\dfrac{7}{5\cdot8}+...+\dfrac{7}{29\cdot32}\right]=\dfrac{3}{4}\)

=>\(x=\dfrac{3}{4}\cdot\left(\dfrac{7}{2\cdot5}+\dfrac{7}{5\cdot8}+...+\dfrac{7}{29\cdot32}\right)\)

=>\(x=\dfrac{3}{4}\cdot\dfrac{7}{3}\left(\dfrac{3}{2\cdot5}+\dfrac{3}{5\cdot8}+...+\dfrac{3}{29\cdot32}\right)=\dfrac{7}{4}\left(\dfrac{1}{2}-\dfrac{1}{5}+\dfrac{1}{5}-\dfrac{1}{8}+...+\dfrac{1}{29}-\dfrac{1}{32}\right)\)

\(\Leftrightarrow x=\dfrac{7}{4}\left(\dfrac{1}{2}-\dfrac{1}{32}\right)=\dfrac{7}{4}\cdot\dfrac{15}{32}=\dfrac{105}{128}\)

hơi khó 

1) 100 - 7 x ( x - 5 ) = 65

    -7 x ( x - 5 ) = 65 - 100

    -7 x ( x - 5 ) = -35

      x - 5 = \(\dfrac{-35}{-7}\)

      x - 5 = 5 

      x = 10 

2) 7 + 2 x ( x - 3 ) =11 

    2 x ( x - 3 ) = 11 - 7 

    2 x ( x - 3 ) = 4

     x - 3  = \(\dfrac{4}{2}\)

    x - 3 = 2

    x = 5

\(2\cdot4\cdot6\cdot8\cdot10\cdot12⋮5\)

\(40⋮5\)

Do đó: \(A=2\cdot4\cdot6\cdot8\cdot10\cdot12+40⋮5\)

\(2\cdot4\cdot6\cdot8\cdot10\cdot12⋮8;40⋮8\)

Do đó: \(A=2\cdot4\cdot6\cdot8\cdot10\cdot12+40⋮8\)

`A = 2.4.6.8.10.12 + 40`

Ta có: 

`2.4.6.8.10.12` có thừa số `8` và `5 `

`=> 2.4.6.8.10.12⋮ 8` và `5`

`40 ⋮ 8` và `5`

`=> A = 2.4.6.8.10.12 + 40 ⋮ 8` và `5 (dpcm)`

-------------------------------

Nếu `a ⋮c` và `b ⋮c  => a + b  ⋮c `

\(\left(4x+12\right)\left(x+5\right)=0\\ \Rightarrow\left[{}\begin{matrix}4x+12=0\\x+5=0\end{matrix}\right.\\ \Rightarrow\left[{}\begin{matrix}4x=-12\\x=-5\end{matrix}\right.\\ \Rightarrow\left[{}\begin{matrix}x=-3\\x=-5\end{matrix}\right.\)

Vậy: \(x\in\left\{-3;-5\right\}\)

(4x+12)(x+5)=0

=>4(x+3)(x+5)=0

=>(x+3)(x+5)=0

=>\(\left[{}\begin{matrix}x+3=0\\x+5=0\end{matrix}\right.\Leftrightarrow\left[{}\begin{matrix}x=-3\\x=-5\end{matrix}\right.\)

\(B=\dfrac{1}{3}+\dfrac{1}{3^2}+\dfrac{1}{3^3}+...+\dfrac{1}{3^{99}}+\dfrac{1}{3^{100}}\\ \Rightarrow3B=1+\dfrac{1}{3}+\dfrac{1}{3^2}+...+\dfrac{1}{3^{98}}+\dfrac{1}{3^{99}}\\ 3B-B=\left(1+\dfrac{1}{3}+\dfrac{1}{3^2}+...+\dfrac{1}{3^{98}}+\dfrac{1}{3^{99}}\right)-\left(\dfrac{1}{3}+\dfrac{1}{3^2}+\dfrac{1}{3^3}+...+\dfrac{1}{3^{99}}+\dfrac{1}{3^{100}}\right)\)

\(\Rightarrow2B=1-\dfrac{1}{3^{100}}< 1\\ \Rightarrow B< \dfrac{1}{2}< 1\left(DPCM\right)\)

Ta có:

\(B=\dfrac{1}{3}+\dfrac{1}{3^2}+...+\dfrac{1}{3^{100}}\\ \Rightarrow3B=1+\dfrac{1}{3}+...+\dfrac{1}{3^{99}}\\ \Rightarrow3B-B=\left(1+\dfrac{1}{3}+...+\dfrac{1}{3^{99}}\right)-\left(\dfrac{1}{3}+\dfrac{1}{3^2}+...+\dfrac{1}{3^{100}}\right)\\ \Rightarrow2B=1-\dfrac{1}{3^{100}}\\ \Rightarrow B=\dfrac{1-\dfrac{1}{3^{100}}}{2}\)

Vì \(1-\dfrac{1}{3^{100}}< 1\) nên:

\(\dfrac{1-\dfrac{1}{3^{100}}}{2}< \dfrac{1}{2}< 1\) hay \(B< 1\)

Vậy...

5: a+b=7

=>a=7-b

b+c=9

=>c=9-b

c+a=8

=>7-b+9-b=8

=>16-2b=8

=>2b=16-8=8

=>b=4

=>a=7-4=3;c=9-4=5

1: a+b=17

a+b+c=20

=>c=20-17=3

a+c=15

=>a=15-c=15-3=1

b=17-a=17-1=16

2: \(\left\{{}\begin{matrix}a+b=5\\b+c=9\\a+c=6\end{matrix}\right.\Leftrightarrow\left\{{}\begin{matrix}a=5-b\\c=9-b\\5-b+9-b=6\end{matrix}\right.\)

=>\(\left\{{}\begin{matrix}14-2b=6\\a=5-b\\c=9-b\end{matrix}\right.\Leftrightarrow\left\{{}\begin{matrix}2b=8\\a=5-b\\c=9-b\end{matrix}\right.\Leftrightarrow\left\{{}\begin{matrix}b=4\\a=5-4=1\\c=9-4=5\end{matrix}\right.\)

3: \(c=\dfrac{abc}{ab}=\dfrac{288}{24}=12\)

bc=96

=>b=96/12=8

\(a=\dfrac{24}{b}=\dfrac{24}{8}=3\)

4: \(\left\{{}\begin{matrix}ab=36\\bc=45\\ca=20\end{matrix}\right.\Leftrightarrow\left\{{}\begin{matrix}a=\dfrac{36}{b}\\c=\dfrac{45}{b}\\\dfrac{36}{b}\cdot\dfrac{45}{b}=20\end{matrix}\right.\)

=>\(\left\{{}\begin{matrix}b^2=36\cdot45:20=81\\a=\dfrac{36}{b}\\c=\dfrac{45}{b}\end{matrix}\right.\Leftrightarrow\left\{{}\begin{matrix}b\in\left\{9;-9\right\}\\a=\dfrac{36}{b}\\c=\dfrac{45}{b}\end{matrix}\right.\)

TH1: b=9

\(a=\dfrac{36}{9}=4;c=\dfrac{45}{b}=\dfrac{45}{9}=5\)

TH2: b=-9

=>\(a=\dfrac{36}{-9}=-4;c=\dfrac{45}{-9}=-5\)