Determine how many shares will be bought back and choose the nearest option below.

The company wants to distribute $90\%$ of the earnings available to common shareholders.
Earnings available to common shareholders = R5 000 000.

$$\text{Amount for repurchase} = 0.90 \times \text{R}5,000,000$$ $$\text{Amount for repurchase} = \text{R}4,500,000$$

The current share price is R5 per share.

$$\text{Number of shares to buy back} = \frac{\text{Amount for repurchase}}{\text{Share price}}$$ $$\text{Number of shares to buy back} = \frac{\text{R}4,500,000}{\text{R}5 \text{ per share}}$$ $$\text{Number of shares to buy back} = 900,000 \text{ shares}$$

The company has 10 000 000 shares outstanding.
The largest shareholder holds 4 550 001 shares, and none of these will be bought back.
This means the shares available for repurchase from other shareholders are:

$$\text{Shares available from others} = \text{Total shares outstanding} - \text{Largest shareholder's shares}$$ $$\text{Shares available from others} = 10,000,000 - 4,550,001$$ $$\text{Shares available from others} = 5,449,999 \text{ shares}$$

Since the calculated number of shares to buy back (900,000) is less than the shares available from other shareholders (5,449,999), the company can proceed with buying back 900,000 shares.

The final answer is $\text{a. 900 000}$.