Prove q-ary representation of n*q

[Robb]

Homework Statement

Let $q \geq 2$ be an integer. Let $n = (d_k d_{k-1} \dots d_1 )_(q)$ be a q-ary representation of n. Prove that $nq = (d_k d_{k-1} \dots d_1 0)_(q)$.

Relevant Equations

as above for n and nq.

$nq = q(d_1 + d_2 q + d_3 q^2 + \dots + d_k q^{k-1})$
$= d_1 q + d_2 q^2 + d_3 q^3 + \dots + d_k q^k$
$= d_k d_{k-1} \dots d_1 0$
Can someone please explain how to get from line two to line three. This is instructors solution and not sure I understand. Thanks!

 

[fresh_42]

In a q-ary representation, we have the lowest digit for $q^0$ and we have to note it. You can always make easy examples with $q=10$ and any numbers you want.