This problem is not as hard as it seems

Relevant wiki: Zero Product Property

I have a neat solution to this problem. Just add 1 to each and every term on the left side. To keep balance, add 4 to the right side. Simplifying will yield that $x=\boxed{416}$ .

$\begin{array} { l c l } \dfrac {315 - x} {101} + \dfrac {313 - x} {103} + \dfrac {311 - x} {105} + \dfrac {309 - x} {107} &=& -4 \\ \left(\dfrac {315 - x} {101} +1\right)+ \left(\dfrac {313 - x} {103} +1\right)+ \left(\dfrac {311 - x} {105} +1\right)+\left( \dfrac {309 - x} {107}+1\right) &= &-4 +4 \\ \dfrac {416 - x} {101} + \dfrac {416 - x} {103} + \dfrac {416 - x} {105} + \dfrac {416 - x} {107} &=& 0 \\ (416 - x) \left( \dfrac1{101} + \dfrac1{103} + \dfrac1{105} + \dfrac1{107} \right) &=& 0 \\ \end{array}$

By zero product property ,

$416 - x = 0 \qquad \Rightarrow \qquad \large \boxed{x = 416} .$

$-1+-1+-1+-1=-4$ so just set the first term equal to -1 and solve for x $\frac{315-x}{101}=-1$ $-101=315-x$ $x-315=101$ $x=315+101=416$

Just try for 315 - x = -101 first,

x = 416;

-1 - 1 - 1 - 1 = 4 is found true with x = 416.

I find the longest way x=1870908416/4497376=416, i never thought other simpilified method by adding 4 in left equation

[SOLUTION 1]
let (315-x)/101 = -1 => x=416
then for x=416, (313-x)/103 = -1, (311-x)/105 = -1 and , (309-x)/107 = -1 therefore [x=416].
[SOLUTION 2]
if we see that
315 + 101 = 416
313 + 103 = 416
311 + 105 = 416
and
309 + 107 = 416
this implies that x=416