"All limits, especially national ones, are contrary to the nature of mathematics… Mathematics knows no races… For mathematics the whole cultural world is a single country" – David Hilbert. "Face problems with a minimum of blind calculation, a maximum of seeing thought" – Hermann Minkowski

How to write proofs by induction about divisibility – video clip

Want to prove that m | f(k+1) if m | f(k)? Then start from

m | f(k)

and fiddle with the right-hand side of that divisibility statement, which you can do either by adding (or subtracting) multiples of m, or multiplying f(k) by any number, until you get