My mom recently died and I am named beneficiary on her checking and savings accounts. I know she would want her other two children to have an equal share (don't understand why she only named me--she loved us all equally). Anyway, when I obtain the money can I write them each a check? Will they have to pay income tax?
You certainly can write each of your siblings a check if that is what you want to do. However, in light of the fact that what you do and your timing could have tax implications, you (and they) speak with a tax advisor before you do this.
Your question about taxes depends on amounts, timing, and each individual's tax position, etc. Talk to a tax advisor for that information.
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