There is no magic way to get someone's social security number. All I can do is give you ideas of where to look---and this is going to require some effort on your part unfortunately.
1. Look at the last joint tax return the two of you filed--state or federal and it will be there.
2. Contact your bank because every bank account you have had in the past with him has both of you social security numbers attached to it.
3. Every credit card the two of you applied for had an application and that application needed your social security card so that you could apply for credit.
4. If you rented an apartment or home, chances are good that you provided a written document to the landlord before you rented with your social security number so that they could run a credit check on you. Look the old landlord up.
5. Any employer that he worked for in since 1936 would have a record of his social security number.
6. Finally, even the military can provide you with a number if you can provide basic information about the service member which is retrievable from his birth certificate, and even his death certificate.
If all else fails, try asking him personally. If you are simply making an ex spouse claim, your claim will IN NO WAY negatively effect your ex.
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I wish you and yours the best in 2016,