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@alayna - That is true, each person qualifying for HOH would need a different child.  However, that is not the subject of this question, which was "Can the custodial parent claim HOH and the non-custodial parent claim the child's exemption?" and the answer to that is - yes (as explained in the answer above).
**Disclaimer: This post is for discussion purposes only and is NOT tax advice. The author takes no responsibility for the accuracy of any information in this post.**