Calculation of a value for number divisible by 11

Question

Calculates the value $x$ for the number $M=5278x$ is divisible by 11

my attempt,

$11\mid M=5278x \Longleftrightarrow (5-2)+(7-8)+x=2+x$ is multiple of $11$

$ 2+x$ is multiple of $11 \Longleftrightarrow 11\mid 2+x \Longleftrightarrow \exists k\in \mathbb{Z} : 2+x=k.11 \Longleftrightarrow x=k.11-2 \text{ with } k\in \mathbb{Z} $

Am i right ?

Answer 1

First, I'm assuming "$5278x$" means "$52780+x$," based on how you manipulated this nonstandard notation. (Normally that would mean multiplication, and $x=0$ or $x=11$ would be sufficient to have $M$ divisible by $11$).

With your argument, you haven't quite finished. You need to actually give a value of $x$ from $0$ to $9$. When $k=1$, you have $x=11-2=9$, and you can check that $52789$ is in fact divisible by $11$. Furthermore, you can see this is the only possibility because $11\cdot k-2$ is greater than $9$ when $k\geq 2$ and $11\cdot k-2$ is less than zero when $k\leq 0$.