Son in 18, graduated from high school Spring 2016, lived with us and we supported him until in enlisted in the Army 10/16, then mailed 12/16. Can we claim him?

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Yes a claimable dependent isn't like health insurance. If he lived with you and you supported him for a minimum of 6 months during the year, you can claim him as a dependent.

Keep in mind you must notify him of your intentions. Since he was receiving income during basic training he will be getting a w-2 from that time to ensure full complaince under tax law when he files.