You must have the address on your residence card changed within 14 days of moving.

[Procedure flow]

Go to the ward/city office of your old address, submit a "tenshutsu todoke", (notification of moving out) and obtain a "tenshutsu shomeisho" (change of address certificate).
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Within 14 days of moving in, go to the ward/city office of your new address and submit the "tenshutsu shomeisho" and "tennyu todoke" (notification of moving in) to register as a resident.
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When the ward/city office changes the address on the back of your residence card to your new address, the process is completed.

*Be sure to change the address on your student ID as well. If you do not know how, inquire your affiliated school office.