If ${\log }_{10}\sin \mathrm{x}+{\log }_{10}\cos \mathrm{x}=-1\text{ and }{\log }_{10}(\sin \mathrm{x}+\cos \mathrm{x})=\frac{\left({\log }_{10}\mathrm{n}\right)-1}{2}$, then the value of ‘n/3’ is ___

Given ${\log }_{10}\left(\frac{\sin 2\mathrm{x}}{2}\right)=-1\text{ or }\frac{\sin 2\mathrm{x}}{2}=\frac{1}{10}\text{ or }\sin 2\mathrm{x}=\frac{1}{5}$

Also ${\log }_{10}(\sin \mathrm{x}+\cos \mathrm{x})=\frac{{\log }_{10}\left(\frac{\mathrm{n}}{10}\right)}{2}$

or $1+\sin 2\mathrm{x}=\frac{\mathrm{n}}{10}\text{ or }1+\frac{1}{5}=\frac{\mathrm{n}}{10}\text{ or }\frac{6}{5}=\frac{\mathrm{n}}{10}\text{ or }\frac{\mathrm{n}}{3}=4$