If custodial parent and child lived with non custodial parent and non custodial parent is the only one that worked and supported both, can the non custodial parent claim child on their tax return?

Yes - as long as the custodian parent agrees. The custiodian parent must sign the waiver IRS form 8332 and give it to the person who is claiming the child to attach to his/her tax return.

Please stop asking the same or partial question over and over.

Also do not ask the same question in two places.

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