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As far as tax purposes go, you can claim a child as a dependent as long as the following applies:
1.) The child is your son or daughter (or other descendant).
2.) The child is under the age of 19 (or 24 if a full-time student) or disabled.
3.) You lived with the child for more than half the year, or the parent with whom the child did live more than half the year signs form 8332 releasing the exemption to you.
4.) The child did not provide more than half of their own financial support.
Most likely, you can claim the child for tax purposes.
The only problem that I could see arising with Medicaid is if your girlfriend did not disclose a.) child support that may have paid by you, b.) whether you live with her and the child, and/or c.) money you pay to support her and the child. Medicaid eligibility is based on household income and taxable and non-taxable income
If anything was not disclosed on her Medicaid application, the child not being claimed on her return may trigger some questions from the Medicaid department. However, I am speculating on possible issues and the chances of Medicaid taking the actions I have mentioned may be pretty slim as well.
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