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I will leap in here. You want to look at the instructions for Form 1040, specifically for the Child Tax Credit on Page 31 and the Earned Income Credit on Page 36.

Let me hypothesize. Suppose the child of divorced parents is under 17, and the mother has custody and pays more than half the cost of keeping up a home for the child. Then she qualifies as Head of Household. However, the father claims the child as a dependant because he is in a higher tax bracket, and it pays to do that. There is some written ageement between the ex-spouses that that is OK.

If the mother doesn't earn all that much, she qualifies for the Earned Income Credit (see Page 36). But it is the father who might be able to claim the Child Tax credit, because it is he who is claiming the child as a dependant (see Page 31). This credit too depends on income, so everybody has to go through the relevant worksheets.

If the mother is claiming the child as a dependant, then it is clear, she gets both credits subject to the income limitations.
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