Mathematical Induction Divisibility Proof (1 of 3: What is inductive logic?)

[SoumilSahu]

If more teachers like you existed, people wouldn't laugh and say "haha i suck at math" but rather proudly say "i use math to improve the world"

[bouncycrabboomz]

infact, 7^a - 4^b would always be divisible by 3 given any positive integer a, b.
This is because 7 is congruent to 1 mod 3, and so is 4.
i.e. 7 to the power of anything would always give you a remainder of 1 when divided by 3,
and 4 to the power of anything would also give you a remainder of1 when divided by 3.
Hence 1-1 the remainder is always 0

This was my inductive step:

=7×7^n-64×4^n-7^n+16×4^n
=6×7^n-48×4^n
Why make it so complicated? You just need to start with f(n+1)-f(n) and simplify. As far as I know, this even works with ALL divisibility proofs, unlike my "increasing gaps/factors" trick for inequality proofs, which has worked for every example on this channel so far, but might fail in some special cases.

"Some of you at home…" For a moment I thought he was talking to the camera. :D