Equality in Z_n

Question

Let $k \in \mathbb{Z}$. Is the following equality true?

$(kx+k\alpha) \text{mod 1} = k(x+a \text{ mod 1}) \text{ mod1}$?

I have reviewed some number theory books but I don't find how to justify that equality. Any help would be appreciated.

Answer 1

From the definition of modulo: We have that for any $a \in Z$, $a=1*a$, and so $(a \mod{1}) = a$. Thus,

$(kx+ka) = (kx+ka) \mod{1} = kx+ka = k(x+a) = k(x+a \mod{1}) = k(x+a \mod{1}) \mod{1}$