\( \definecolor{colordef}{RGB}{249,49,84} \definecolor{colorprop}{RGB}{18,102,241} \)
Let \(x\) be a natural integer and \(a_k a_{k-1} \dots a_1 a_0\) its decimal representation.
  1. Given that \(10 \equiv -1 \pmod{11}\), show that: $$x \equiv (a_0 + a_2 + a_4 + \dots) - (a_1 + a_3 + a_5 + \dots) \pmod{11}$$
  2. State a divisibility criterion by \(11\).
  3. Determine the remainder in the division by \(11\) for each of the following integers:
    1. \(123\,456\,789\)
    2. \(10\,891\,089\)

Capture an image of your work. AI teacher feedback takes approximately 10 seconds.