Solve {l}{text{If}(m,n)=1text{,provethat}}{qquadsum_x=1^n-1[mx/n]=(m-1)(n-1)/2}

$$\left. \begin{array} { l } { \text { If } ( m , n ) = 1 \text { , prove that } } \\ { \qquad \sum _ { x = 1 } ^ { n - 1 } [ \frac { m x } { n } ] = \frac { ( m - 1 ) ( n - 1 ) } { 2 } } \end{array} \right.$$

Solve for x (complex solution)

$$\left\{\begin{matrix}x=-\frac{n\left(1-n\right)\left(m-1\right)}{2m}\text{, }&m\neq 0\text{ and }n\neq 0\\x\in \mathrm{C}\text{, }&m=0\text{ and }n=1\end{matrix}\right.$$

Steps for Solving Linear Equation

Multiply both sides of the equation by 2n, the least common multiple of n,2.

$$2mx=n\left(m-1\right)\left(n-1\right)$$

Use the distributive property to multiply n by m-1.

$$2mx=\left(nm-n\right)\left(n-1\right)$$

Use the distributive property to multiply nm-n by n-1.

$$2mx=mn^{2}-nm-n^{2}+n$$

The equation is in standard form.

$$2mx=mn^{2}-mn-n^{2}+n$$

Divide both sides by 2m.

$$\frac{2mx}{2m}=\frac{n\left(m-1\right)\left(n-1\right)}{2m}$$

Dividing by 2m undoes the multiplication by 2m.

$$x=\frac{n\left(m-1\right)\left(n-1\right)}{2m}$$

Solve for x

$$\left\{\begin{matrix}x=-\frac{n\left(1-n\right)\left(m-1\right)}{2m}\text{, }&m\neq 0\text{ and }n\neq 0\\x\in \mathrm{R}\text{, }&m=0\text{ and }n=1\end{matrix}\right.$$

Steps for Solving Linear Equation

Multiply both sides of the equation by 2n, the least common multiple of n,2.

$$2mx=n\left(m-1\right)\left(n-1\right)$$

Use the distributive property to multiply n by m-1.

$$2mx=\left(nm-n\right)\left(n-1\right)$$

Use the distributive property to multiply nm-n by n-1.

$$2mx=mn^{2}-nm-n^{2}+n$$

The equation is in standard form.

$$2mx=mn^{2}-mn-n^{2}+n$$

Divide both sides by 2m.

$$\frac{2mx}{2m}=\frac{n\left(m-1\right)\left(n-1\right)}{2m}$$

Dividing by 2m undoes the multiplication by 2m.

$$x=\frac{n\left(m-1\right)\left(n-1\right)}{2m}$$