Smallest Ordinal which satisfies an equation.

Question

How can I find the smallest ordinal $\beta$ which satisfies the equation:

$\omega + \beta = \beta\;$?

Answer 1

HINT: Observe that it also has to satisfy

$$\omega+\omega+\beta=\omega+\beta=\beta\;,$$

$$\omega+\omega+\omega+\beta=\omega+\beta=\beta\;,$$

and in general $\omega\cdot n+\beta=\beta$ for $n\in\omega$. That fact should tell you what $\beta$ has to look like.