Unfortunately, because renting an apartment is a private business transaction, one party (landlord) can ask for information from the other party (potential tenant). If a potential tenant does not want to provide the information, the landlord can say, "I choose not to rent to you."
It is similar to providing your Social Security number. You don't have to provide it for most routine transactions, but if you don't, you might be denied goods or services.
Generally, it is a good rule not to provide information of this sort in this kind of transaction because of the opportunity for identity theft or an attempt to have rent payments automatically deducted from a bank account, etc. (Although an account holder must authorize automatic payments, some creditors may manage to bypass the requirement.)
If the landlord is looking for some assurance of creditworthiness, he or she should ask prospective tenants for a credit report or something that shows accounts in good standing (without providing the account numbers).
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