How do you know when you can apply the induction hypothesis in strong induction??

I understand how to get the base cases.

And then assume that P(n).

Then I understand that I need to prove P(n+1).

What I don't understand is why you are able to apply the induction hypothesis. I know the reason is so that you can augment is to see the truth of P(n+1) but I don't understand when you are able to do so. Thanks

*It would be helpful to show me via example of proving the existence of the "division algorithm," but anything is helpful. Thanks so much