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Hello and thank you for using Just Answer.You did not ask about this but I am offering the info just incase. You would need to withhold 30% on the Alimony paid to your ex unless their home country has a tax ttreaty with the US that exempts this. You can verify this by looking at IRS Publication 504. Your ex could then file a US return to claim the refund for the withholding. As far as not having the SSN for your ex you are instructed to enter "NRA" in the space for the SSN. This may require that you mail the return instead of being able to e-file for each year that you are using the 8332 to claim the children as dependents. Do not forget that your ex spouse is required to sign the Form 8832.