$($Stock repurchases and earnings per share$)$CareMore$,$ Inc. provides$$ in$-$home medical assistance to the elderly and earned net income of $$4\mathrm{.}6$ million that it plans to use to repurchase shares of the$$ firm's common$$ stock, which is currently selling for $$50$ a share. CareMore has $10$ million shares of stock outstanding.

a$.$ What fraction of the$$ firm's shares can the firm repurchase for $$4\mathrm{.}6$million$?$

b$.$ If the share repurchase has no impact on the$$ firm's net$$ income, what will be its earnings per share after the$$ repurchase?