Puzzle #37 What is wrong with this argument?

Let A = 1.

Then A² = 1.

So A² - a = 1 – 1 = 0.

But A²- A = A(A – 1).

So A(A – 1) = 0.

Now divide both sides of this equation by A - 1 to get

A(A – 1)/(A – 1) = 0/(A – 1),

which is the same as A = 0, since 0 divided by any whole number is 0.

Thus 1 = 0 ?

What is wrong with this argument?