Time machine
have a nerw MacBook Air and time machine hard drive used with previous MacBook Air now cannot get time machine to backup Do I format drive? or?
MacBook Air 13″, macOS 12.3
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If it's not functioning yes, it should be reformatted and your backups will start again when you launch Time Machine after you select the new drive - (see Back up your Mac with Time Machine). Some people have the practice of doing this anyway when they acquire a new Mac. I've sometimes found that 'inheriting' a backup regime on a new Mac has problems and I have to reformat the drive.
See Erase and reformat a storage device in Disk Utility on Mac