Does paying your rent within that grace period effect Credit score?

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example: i have no credit score, so when i moved in with my roommate's apartment it was perfect because her name was all ready was on the lease. I could have put my name on the lease but i didnt know how responsible she was when it came to paying rent, so i decided not to. I want to pay our rent every 1st of the month, but she likes to wait till the last day we are allowed to pay our rent to pay our rent. Previously i said nothing because its not affecting me. BUT NOW we are trying to renew our lease and that means both of our names are going to be on the lease. My question is that if we continue paying on the last day of the grace period will it affect our credit score? i dont want to start out my credit score history rocky.

Does paying your rent within that grace period effect Credit score?