Which of the following is a necessary condition after a company repurchases shares?

A company’s share repurchase, also known as a buyback, involves the company buying back its own shares from the marketplace, which can be for various strategic reasons, including returning capital to shareholders or reducing the number of outstanding shares.

Having at least one non-redeemable share remaining in issue is critical. This is due to legal and regulatory requirements that prevent a company from becoming entirely owned by itself, which could create issues related to solvency, shareholder rights, and corporate governance. If a company were allowed to repurchase all its shares, it would effectively be eliminating its ownership structure and potentially limiting the ability of any remaining or future shareholders to have a stake in the company.

While maintaining records of shares bought back is essential for proper accounting and regulatory compliance, it is not a condition that affects the fundamental nature of the company's ability to operate post-repurchase. An increase in share value or requiring all shareholders to approve the repurchase are not mandatory conditions either; a company may repurchase shares without needing unanimous shareholder approval, depending on the applicable laws and corporate bylaws.

Thus, ensuring that at least one non-redeemable share remains in issue is essential for the legality and ongoing operation of the company following a repurchase.