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If the loan is joint and several as to all partners, one partner has the right to pay off the loan and seek reimbursement from the other partners. It is likely that the company's operating agreement states something to this effect with regard to profit and loss sharing or about debts incurred by the company.
Unless the corporate by laws state something different, the partner that "bails out" the others has a right to seek contribution and repayment from the others.
If the operating agreement is silent on this matter, then you would have to look to the state statute of your state dealing with corporations/llc's/partnerships, or even just basic contract law. I'm not aware of a state that does not give a joint debtor the right to recover money paid on behalf of another.