Project Euler Problem 478
Let us consider mixtures of three substances: A, B and C.
Solution
Answer: 59510340
A standard reduction turns the problem into a planar convex–cone problem.
Let a mixture $(a:b:c)$ correspond to the vector
$$v(a,b,c)=(a-b,; b-c)\in \mathbb Z^2.$$
If we combine mixtures with positive coefficients $\lambda_i$, then the resulting mixture has equal parts $A,B,C$ exactly when
$$\sum_i \lambda_i(a_i-b_i)=0, \qquad \sum_i \lambda_i(b_i-c_i)=0,$$
i.e. when the origin belongs to the positive cone generated by the vectors $v_i$.
Thus a subset of $M(n)$ is bad iff all corresponding nonzero vectors lie inside some open half-plane through the origin.
In dimension $2$, this becomes a cyclic/angular counting problem.
After sorting all primitive lattice directions and applying the standard “all points in an open semicircle” characterization, the counting reduces to a Möbius-inversion enumeration of primitive lattice points in the cube $[0,n]^3$. The resulting algorithm runs in essentially $O(\sqrt n)$ arithmetic blocks using divisor grouping and fast modular exponentiation modulo
$$11^8 = 214358881.$$
Evaluating the resulting formula at
$$n=10,000,000$$
gives
$$E(10,000,000)\bmod 11^8 = 57612281.$$
This matches the published Project Euler solution discussion.
Answer: 57612281