The AC-method is often presented as an unmotivated rule-based algorithm, e.g. here. If that is the way that you learned it, then I can understand why you hate it. Below I explain the (little-known) ...

The problem is due to imprecise specification of the intended result. Here's a more precise way. $\text{Recall that the }{\bf commutative\ law}\ \color{#c00}X+ \color{#0a0}Y = Y + X\ \text{ is true ... View answer 9 votes Tiling problems might meet your constraints. A nice simple example is Golomb's Theorem that a chessboard of side$2^n$with any square omitted can be tiled by trominoes ("L" shapes of 3 squares). In ... View answer Accepted answer 9 votes Both are valid ways to prove the uniqueness of solutions, i.e. that the set$\,S\,$of solutions contains at most one element. Indeed, one easily proves that the following are equivalent for any set$\...

Some nice geometrical applications arise in the analysis of periodical curves such as Roulettes (Spirograph curves), Star Polygons, etc. Concrete experience with implementations in toys like ...

Proving equivalent the below characterizations of squarefree naturals is quite elementary employing cyclic inferences (assuming one knows the existence and uniqueness of prime factorizations). ...

One way to avoid (explicit) division is to make a change of change variables $\, X = x-a\,$ which reduces it to the following simpler special case $$X\mid P(X) \iff \color{#c00}{P(0)} = 0$$ ...

Quartics and higher degree polynomials frequently arise when intersecting lower degree curves, e,g, the intersection of two conics, or a line and a torus. Such curve intersections often occur in ...

[Note: I interpreted the question as how to explain the relationship between the two definitions] Let's first consider the underlying algebra before turning to real-world models. As an example, ...

Below are a various methods of presenting fraction inversion based on various innate symmetries. Unlike some other methods, these ideas are more to the heart of the matter, so they generalize more ...

A nice third problem arises by continuing from the second problem to show the power of parity arithmetic for solving problems in integer arithmetic. For example, we can use it to show that large ...

Perhaps worth emphasis is that the use of prime factorizations (either directly or implicitly via (multiset) Venn diagrams or exponent vectors) may actually obfusctate the essence of the matter. For ...

I won't try to give a specific explanation here because I think that depends highly on the level of knowledge of the student. Rather, I'd like to emphasize some general points that I think are helpful ...

There is no standard form for such expressions. However, some forms may prove more convenient for certain applications, e.g. transforming the radicands to be squarefree helps to ascertain ...

Often students wrongly believe that integer factorization is unique because there is a deterministic algorithm for factorization: simply continue to pull out the least prime factor (found by trial ...

Telescopy (or telescopic induction) is a nice example. It is a simple, prototypical example of induction that will provide strong intuition for later studies. Telescopic cancellation is usually ...

Update. It appears that the intent of this answer was not clear to some readers, so I will elaborate. The most common way to fix the student's computations is simply to break out the subexpressions ...

I recommend choosing an elementary proof that includes both elements of surprise and beauty - one that might spark students to further study mathematics. Along these lines I've had much success ...

First, we find an integer universally coprime to any prime $\,p.\,$ A moments thought yields the solution $\color{#c00}1$. Next, we generalize to larger solutions. Since the coprimality of $\,n\,$ to ...