There are two video clips here. The first gives one way of doing Step 2 in proofs by induction about divisibility.
Want to prove that m | f(k+1) if m | f(k)? Then “anti-simplify” f(k+1) until you get:
m = something × f(k) + something else divisible by m
I prefer the way in the second clip, where we go the other direction, starting from
m | f(k)
and manipulating that equation until it tells us that
m | f(k+1)
Whichever you prefer.
This is the first way:
And here’s the other way (the better one, I think). This video is a bit longer, because it goes through four different examples.