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You are allowed for a partial credit on the homestead rental as long as you lived there for 6 months or longer.
There is also the income calculation. Part year residents must annualize household income to determine if a credit reduction applies.
When you rent, use total rent paid, then prorate the first/last month based on days of occupancy to determine the total amount of rent that may be claimed for credit.
Your credit would be based on your MI income. Check your MI-1040CR, Part 3, to determine the taxes eligible to be claimed for credit on your Michigan homestead.