In this case no, you are not eligible for the Other State Tax Credit. In your situation, you are filing part-year returns for both Georgia and Illinois. Georgia taxes the income you had over there, and Illinois taxes the income you earned in that state. Neither is taxing the other state's income.
The Other State Tax Credit is needed if you live in one state and work in another at the same time. This is because both states are taxing the same income. However, to reduce the effect of double-taxation, the resident state will usually give you a credit for taxes you pay to the nonresident state on the income earned in that state.
But in your case, only Georgia taxes the Georgia income and only Illinois taxes the Illinois income, so the credit is not used.
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