Core/shell nanowire (NW) is recognized as promising one-dimensional material for nanoelectronic and nanoelectromechanical systems. However, its mechanical properties so important for engineering applications remain largely unexplored. Based on the density functional theory (DFT), we theoretically investigate mechanical and electronic properties of the Ge-core/Si-shell NWs along the [100] direction within the cross sectional size of 1.0 nm and 1.4 nm under the axial strain. Our results show that ideal strength of Ge-core/Si-shell NWs strongly depends on wire cross sectional size compared with that of the Si and Ge NWs. Ideal strength (maximum tensile strength) of Ge-core/Si-shell NWs increases significantly when increasing thickness of the Si-shell. We found that bond lengths around interfaces between the core and the shell play a predominant role in ideal strength of Ge-core/Si-shell NWs. Additionally, band structures of NWs are modififed by applying axial strain. Band gaps of NWs decrease with increasing strain. Our results provide important insight into intrinsic mechanical behavior and electronic properties of Ge-core/Si-shell NWs, useful for the design of nanodevices with Ge-core/Si-shell NWs in future applications.