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You are using a rather old version of npm. The current version is 6.9.0, and the packages you are working with are asking for npm >= 5.5.1.
You need to update your node version, if possible always use latest node version
Nodejs version 14.4.
And if you have project that required different node version then use
NVM - node version manager
Just use sudo for global install
sudo npm install -g @vue/cli
Source:stackexchange.com