Convert all to base 10\[\begin{align}  & {{256}_{7}}=2\times {{7}^{2}}+5\times {{7}^{1}}\times 6\times {{7}^{0}} \\ & =98+35+6=139 \\ & {{314}_{10}}-{{139}_{10}}={{175}_{10}} \\ & {{175}_{10}}={{340}_{x}} \\ & 175=3\times {{x}^{2}}+4\times {{x}^{1}}+0\times {{x}^{0}} \\ & 175=3{{x}^{2}}+4x \\ & 3{{x}^{2}}+4x-175=0 \\ & 3{{x}^{2}}+25x-21x-175=0 \\ & x(3x-25)-7(3x-25)=0 \\ & (x-7)(3x-25)=0 \\ & x=7\text{ or }x=\tfrac{25}{3} \end{align}\]Ignore the fractional part, x =7