We have a big solution with thousands of warnings. Would it take less to compile the solution if I removed all of the warnings (either manually or using a tool)?
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You should not silent the warnings. Most of the warnings indicates problems with your code. You should look into your code and fix them.
Processing all the warning should cost compile time (The compieler need to send the warnings to visual studio, more warnings should mean more output data. But I don't know how big the impact is). But if you fix the warnings you propably fix bugs, too.