Deductions & credits

It depends if she is the custodial parent.  While you can agree to allow the other parent to claim the child as a dependent, only the custodial parent (one the child spends more nights with) can claim the child for Earned Income Credit (EIC), Child and Dependent Care credit (CDC), and Head of Household.  You cannot "shift" those via agreement.  You can, however, shift the dependency exemption and the Child Tax Credit.

So, when you are answering the "how many months" question, the program is trying to determine if you are the custodial parent (and thus who gets the EIC etc and why your refund is much larger if over 7 months).  If the time spent is relatively even and you aren't sure how to count the months, here is a blurb from the TurboTax program that explains this -

"Sometimes this is a bit complicated. If you find yourself saying "a little over 6 months," then you need to count the exact number of days this person lived with you. If it was fewer than 183 days (184 in a leap year), then round down to 6 months. [if more, then round up to 7 months]

Example: Sally's son Jack lived with her for just over 6 months, but it wasn't for more than half of the total number of days in the year, which is 183 days (184 in a leap year). That means Sally should enter 6 months.

Otherwise, you can count a partial month as a full month if they lived with you for less than 6 months or more than 7 months."

Assuming your ex-wife is the custodial parent, then she will enter 7 months and proceed through the questions.  After entering dependent information, there will be a series of questions that asks if she has an agreement with the other parent and will allow them to claim the dependent.  By answering yes to those, you will be able to claim the child as a dependent and for the Child Tax Credit.  But, she will still retain the dependent for EIC/CDC/HOH.

But, there is more.  While perfectly acceptable tax law wise, this scenario can cause problems with the e-file system because the child's SSN remains on both returns.  What will happen is the IRS e-file system is hyper sensitive (due to general ID protection measures) and it will still read the child's SSN on both returns.  And, thus, it will reject either your return or the other return, whichever is filed second.

The solution is for one party to paper file their return.  You can try to e-file both but you will likely receive a rejection when you file, so I want to make you aware of this now, rather than later.