Inequality with sum of divisors ending in a specific digit

[tall ore]

1. Ask your question and show the work you've done so far. If you've posted a screenshot of a question, specify which part you need help with.
2. Wait patiently for a helper to come along.
3. Once someone helps you, say thank you and close the thread with:

   +close
   

4. Feel free to nominate the person for helper of the week in #helper-nominations
5. Do not ping the mods, unless someone is breaking the rules.
6. If you're happy with the help you got here, and the server overall, you can contribute financially as well:

[slender shoal]

I'm pretty stuck on the assignment for my number theory class.. one of which is the following:
For a natural number $n \in \mathbb{N}$ and a digit $j \in {0, \ldots, 9}$ let $\tau_j(n)$ be the amount of positive divisors of $n$, with $j$ as their last (one's) digit. Show that
$$ \tau_3(n) + \tau_7(n) \leq \tau_1(n) + \tau_9(n). $$
There's also a hint given, namely: Show, that $\alpha(n) \coloneqq \tau_1(n) + \tau_9(n) - \tau_3(n) - \tau_7(n)$ is a multiplicative function, but I don't know, how I should prove that hint. 😅
Could someone give me a hint how to approach this kind of problem?

+close