Hello student yes you're right getting admission into NIT and IIT for M.Tech course is through GATE Entrance examination only.
Eligibility - candidates must have passed B.tech /BE with a valid gate Score there's no upper age limit.

You need do register separately for the respective institutes at their respective official website. Also candidates need to register at GATE COAP 2020 Portal.

For more updates refer to the link mentioned below - https://www.google.com/amp/s/engineering.careers360.com/articles/iit-mtech-admissions/amp
Feel free to comment if you have any doubt
Good luck

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