I bought a router(DI-524) to connect the four computers then I bought another two computers but does't have internet connection because only four ports available for this type of router, is it okay if i'll buy another router(DI-524) to connect these another two computers?
The best device that your can use is a switch for your network. The purpose of the switch is to expand a wired devices on your network. No need to use another router since the main router is the one who produces internet and DHCP on your network.
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