When all directors of a company resign simultaneously, it raises questions about who can sign the board resolution to accept these resignations, as a company cannot technically exist without directors. The consensus is that this situation is problematic and could lead to the company's dissolution. It's recommended to consult with a Chartered Accountant (CA) and Company Secretary (CS) for guidance. A practical approach suggested is for directors to resign one by one, allowing the remaining director to appoint new ones before their own departure.