Generalize the argument of Exercise 39 to prove that \(\sqrt{A}\) is irrational if \(A\) is a whole number but not a perfect square. Choose \(n\) as before and let \(m=n \sqrt{A}-n\lfloor\sqrt{A}floor\), where \(\lfloor xfloor\) is the greatest integer function.

Data From Exercise 39

Carry out the details of the following proof by contradiction that \(\sqrt{2}\) is irrational (this proof is due to R. Palais). If \(\sqrt{2}\) is rational, then \(n \sqrt{2}\) is a whole number for some whole number \(n\). Let \(n\) be the smallest such whole number and let \(m=n \sqrt{2}-n\).
(a) Prove that \(m(b) Prove that \(m \sqrt{2}\) is a whole number.
Explain why (a) and (b) imply that \(\sqrt{2}\) is irrational.