Why is the product of both numbers always an integer root?

Question

I have two numbers. $x, p$

This numbers, have a integer root, then: $\sqrt{x} \in \mathbb Z, \sqrt{p} \in \mathbb Z$. And also:

$x = c^2$, $p = d^2$, because it have a integer root.

So, prove that $xp = k$, where $k$ have a integer root.

My development was:

If, $xp = k$ , that is: $c^2d^2 = k$ and since $c,d$ are integer numbers, so:

$cd = \sqrt{k}$, And here I have not been able to continue, because now I need to prove that:

The product of two integers, will always be integer

¿How i can prove it?

Answer 1

Let $a,b \in \mathbb{Z}$. Since the set of integers forms a semigroup under multiplication:

$a,b \in (\mathbb{Z}, *)$ then $a*b \in (\mathbb{Z},*)$ and $b*a \in (\mathbb{Z},*)$ (Satisfies closure).

Note: it is a semigroup because it doesn't satisfy inverse axiom.

Answer 2

This is a really general question about multiplying numbers. If the course is not too basic, it is ok to continue by stating that it is trivial. If you want to have a proof, see http://foolproof.pbworks.com/w/page/8978143/The%20Product%20Of%20Two%20Integers%20Is%20An%20Integer